
310 F.Supp. 94 (1970)
Margaret CAMPBELL et al., Plaintiffs,
v.
The BOARD OF EDUCATION et al., Defendants.
No. 70-C-93.
United States District Court, E. D. New York.
February 11, 1970.
Howard I. Kalodner, New York City, for plaintiffs, Lawrence H. McGaughey, Williamsburg Legal Services, Richard S. Panebianco, Attorney-in-Charge, Arnold Rothbaum, Brooklyn, N. Y., of counsel.
*95 Louis J. Lefkowitz, Atty. Gen. of State of N. Y., New York City, for defendants, Daniel M. Cohen, Asst. Atty. Gen., J. Lee Rankin, Corp. Counsel, Irwin L. Herzog, New York City, of counsel.

MEMORANDUM, ORDER, FINDINGS OF FACT AND CONCLUSIONS OF LAW FOR PURPOSES OF PRELIMINARY INJUNCTION
WEINSTEIN, District Judge.
In this class action for an injunction against local school board elections in New York City, plaintiffs attack the constitutionality of the State statute under which the defendants, the Boards of Education and Elections, propose to act. The complaint alleges that the system of proportional representation adopted by the State will result in deprivation of equal protection of the laws and denial of due process. Since the constitutionality of a state statute has been drawn into question, notice of this action has been given by order of the Court to the Attorney General of the State of New York. Upon its motion, the State was allowed to intervene as a defendant.
Plaintiffs' claim  a novel one in the federal courts, as they concede  is that the voting plan is illegal because ballots are counted using a system that may, through the operation of chance, aid some candidates at the expense of others or enable some votes to have a greater effect than others. Defendants have moved to dismiss for failure to state a cause of action. Since the challenged statute applies only locally, it is not necessary to convene a three-judge court. 28 U.S.C. § 2281; Moody v. Flowers, 387 U.S. 97, 87 S.Ct. 1544, 18 L.Ed.2d 643 (1967). For the reasons below, and because the plaintiffs have failed to make "a clear showing of probable success" (Clairol Inc. v. Gillette Co., 389 F.2d 264, 265 (2d Cir.1968)), the motion for a preliminary injunction must be denied.
There are still a number of unresolved questions of fact; the statistical and other data underlying this dispute were not developed at the hearing on the motion for a preliminary injunction with clarity sufficient to warrant a final judgment. Decision on motions to dismiss and for summary judgment should be held in abeyance to permit the parties to gather and present further evidence should they choose to do so.
Because of the imminence of elections, we assume that plaintiffs will wish to take an immediate interlocutory appeal pursuant to section 1292(a) (1) of Title 28 of the United States Code from this order refusing the preliminary injunction. The posture of the case has been set out below in some detail in order to render as much assistance as possible to the Court of Appeals and to the parties. The conclusions are tentative, made only for the purpose of deciding the motions before us.

I. BACKGROUND
The New York State Legislature, unlike Congress, often provides scanty legislative history. Because of this, in determining the factual basis for its actions in any litigation challenging the constitutionality of one of its laws, extensive resort to judicial notice is required to appraise the factual basis for its action, See, e.g., Turner v. United States, 396 U.S. 398, 90 S.Ct. 642, 24 L. Ed.2d 610 (1969); Leary v. United States, 395 U.S. 6, 38, 89 S.Ct. 1532, 23 L.Ed.2d 57 (1969); Kramer v. Union Free School District No. 15, 282 F.Supp. 70, 76 (1968) (dissent), reversed 395 U.S. 621, 89 S.Ct. 1886, 23 L.Ed.2d 583 (1969); Karst, Legislative Facts in Constitutional Litigation, 1960 Sup.Ct. Rev. 75, 77, 84; Alfange, The Relevance of Legislative Facts in Constitutional Law, 114 U. of Pa. L.Rev. 637 (1966). Accordingly, we turn to the facts as the New York Legislature might have viewed them.

A. Educational Problems in New York City.

This Court is no stranger to the controversies that, in recent years, have *96 swirled about public education in the City of New York. Knight v. Board of Education, 48 F.R.D. 108, 115 (E. D.N.Y.1969); Oliver v. Donovan, 293 F.Supp. 958 (E.D.N.Y.1968). The history of public education in New York for the past few years has been one of crisis and confrontations, legal actions, threats, claims and denials. See, e. g., M. Berube & M. Gittell, Confrontation at Ocean Hill-Brownsville (1969); P. Sexton, Spanish Harlem (1965). Within this period, extensive studies have been made of the school system, and numerous solutions have been proposed to settle the continuing controversies. See, e. g., Mayor's Advisory Panel on Decentralization of the New York City Schools, Reconnection for Learning (1967); Board of Education, City of New York, Decentralization  Statement of Policy (1967), reprinted in part in M. Berube & M. Gittell, Confrontation at Ocean Hill-Brownsville 17 (1969). In response to demands for reform  largely centered about proposals for more local control of schools, see, e. g., Oliver v. Donovan, 293 F.Supp. 958 (E.D.N.Y. 1968); Oliver v. Board of Education, 306 F.Supp. 1286 (S.D.N.Y.1969)  the State of New York adopted a decentralization program for New York City's schools. L.1969, c. 330, as amended, L.1969, c. 422 (1969), as amended, L.1970, c. 3 (1970). Under this plan local school boards  with much greater autonomy than heretofore existed  were to be created. It is in the context of this history and of the pervasive educational, social and political problems that it reflects that the action of the State must be viewed.

B. Urban Educational Crisis.

New York City  like metropolitan areas throughout the United States  is faced with problems that cut across the entire fabric of urban life. Growth and large population movements have added to increasing frustration and alienation on the part of residents. Lost and isolated among the multitudes of the city, some individuals have come to believe that they have no influence or control over the actions of government most intimately affecting their day-to-day lives. J. Bollens and H. Schmandt, The Metropolis 218-20 (1965); Herson, The Lost World of Municipal Government, in Urban Government 3, 18 (E. Banfield ed. 1961); Warren, Politics in the Ghetto System, in Politics and the Ghettos 11, 28 (R. Warren ed. 1969).
Nowhere in urban life is this feeling of impotence and lack of identity in the face of the huge metropolitan governmental machine more likely to be felt than in the area of public education. Effective control of large educational systems has, some assert, been entrusted to a professional educational elite, effectively responsible to no one outside the confines of the profession. See, e. g., M. Gittell, Participants and Participation 23-31 (1967); Rosenthal, Pedagogues and Power, in Educating an Urban Population 185 (M. Gittell ed. 1967); Eliot, Public School Politics, in Urban Government 515 (E. Banfield ed. 1961). The result of this tendency, some argue, has been the development of what they have termed the "fortress school," one relatively isolated from the community's problems and insufficiently responsive to its needs. See P. Schrag, Village School Downtown 163-64 (1967); P. Sexton, Spanish Harlem 149-50 (1965).
It has been claimed by some that centralized, city-wide administration permits schools to ignore local needs. Centralized agencies, so the argument goes, have been among the causes of the failure of the schools to produce graduates sufficiently trained to function effectively as citizens and participants in urban political and social life. But cf. Cunningham, Models for Model City Management, in Education and Urban Renaissance 95, 97 (R. Campbell, L. Marx & R. Nystrand eds. 1969); Recommendations, in Id. at 133, 136-43 (schools are more efficiently administered than other local agencies but substantial changes are needed). Whatever the cause, studies conducted throughout *97 the nation have underlined the fact that students in central-city schools fail to match the achievements of those attending schools of outlying urban and suburban areas.
"The most comprehensive review of the [New York City school] system in the last ten years was a 1962 report of the State Education Department. Its major conclusions were:
1. Pupil achievement is generally below the rest of the state. A large proportion of high school students fail to meet minimum standards.
2. Major improvements in curriculum are needed, particularly at the elementary and junior high school levels.
3. The caliber and preparation of the teaching staff are very uneven.
4. Classes are too large, with concomitant results that the staff has heavier teaching loads and more housekeeping chores than are educationally desirable.
5. The areas of greatest need often have the poorest and least experienced teachers.
6. Supplementary social and psychological services are inadequate.
7. Many old, unsound school buildings are still in use.
8. The procedures of the Board of Examiners need to be replaced by less time-consuming methods of teacher selection." (Footnote omitted.) Mayor's Advisory Panel on Decentralization of the New York City Schools, Reconnection for Learning 89 (1967). See also, e. g., P. Sexton, Education and Income 27-28, 30, 139-40 (1961) (Detroit); Toward Creating a Model Urban School System 96-99 (A. H. Passow ed. 1967) (Washington, D.C.); Campbell & Shalala, Problems Unsolved, Solutions Untried: The Urban Crisis, in The American Assembly, The States and the Urban Crisis 11-13 Campbell ed. 1970).

C. Decentralization As A Possible Solution.

A remedy that has been frequently suggested for urban school deficiencies is decentralization. See generally Note, Urban School Decentralization: The Problem of the One and the Many, 5 Colum.J.L. & Social Problems 137 (1969). Proponents assert that local community control of schools will result in greater participation as well as greater responsiveness. See Mayor's Panel on Decentralization of the New York City Schools, Reconnection for Learning 15-16 (1967); Toward Creating a Model Urban School System 9-13, 156-60 (A. H. Passow ed. 1967); Brager, Effecting Organizational Change Through a Demonstration Project: The Case of the Schools, in Community Action Against Poverty 185 (G. Brager & F. Purcell eds. 1967); M. Gittell & T. E. Hollander, Six Urban School Districts 51 (1968) ("Of the three functions the most direct and clear cut cause and effect relationship with innovation appears to be public participation."); C. Westby, Local Autonomy for School Communities in Cities (1947). Cf. M. Thomas, Community Governance and the School Board: A Case Study 31 (U. of Texas, Inst. of Pub. Affairs, 1966) (school board members tie schools to all segments of community). But cf. P. Schrag, Village School Downtown 179-82 (1967) (suggesting urban-suburban centralization); J. Gallagher, School District Reorganization: History, Theory and Law 9 (U. of Calif., Davis, Institute of Gov. Affairs, 1966) (same); Committee for Economic Development, Reshaping Government in Metropolitan Areas 17 (1970) ("what is needed is a system of government that adequately recognizes both forces, centralization and decentralization.") (Emphasis in original)
It has been urged that participation and responsiveness are self-reinforcing. That, when parents can control to a significant extent the policies and directions of the local schools, the schools will *98 become more responsive. And, in turn, faced with a more responsive and concerned school system, parents and students may more effectively and easily participate in the educational process. With more community control and more responsiveness, it is said, those less articulate and more self-conscious individuals who have been unable or unwilling to influence the educational process may be drawn within its ambit. See, e. g., P. Sexton, Education and Income 227-29 (1961); P. Sexton, Spanish Harlem 63 (1965); Toward Creating a Model Urban School System 65-72, 156 (A. H. Passow ed. 1967); Dodson, Education and the Powerless, in Education of the Disadvantaged 61, 71-72 (A. H. Passow, M. Goldberg & P. Tannenbaum eds. 1967). Cf. M. Kotler, Neighborhood Government (1969).
In proposing decentralization, experts have suggested many different plans dealing with organization, powers and selection of local units. New York City, for example, has had a form of decentralization for several years although the powers of local boards have been limited. Proposals for advisory boards  appointed or elected  totally autonomous boards, and boards with powers ranging between these extremes have all been made or are in use throughout the United States. See Toward Creating a Model Urban School System 159, 375 (A. H. Passow ed. 1967); M. Gittell & T. E. Hollander, Six Urban School Districts 16-18, 64-76 (1968); Mayor's Advisory Panel on Decentralization of the New York City Schools, Reconnection for Learning (1967). Though substantial agreement exists among many students of the subject that some form of decentralization is desirable, it is true that:
"Decentralization * * * is still basically a grey area with very few dependable guidelines as large city systems find when they grapple with the problem. For the moment, decentralization is a label for a complex set of concepts for policy and administrative decision making * * *. There are hard questions concerning the units of a decentralized system; the boundary lines; the extent to which a unit should be a real community rather than just a geographical area * * *. Compounding the problem * * * is the need, in the process of decentralization, to build a sense of polity in subcommunities where there has been little experience in participating as the normal citizen would do in any other community in the nation." Toward Creating a Model Urban School System 167-70 (A. H. Passow ed. 1967).
Whatever the merits of the arguments, it is apparent that New York State's Legislature credited those which support decentralization when it devised the plan now being challenged by plaintiffs.

II. STATUTORY SCHEME
In addition to increasing the authority of local school boards in New York City, New York State has chosen proportional representation as the procedure for electing members of these boards. The procedure is outlined in section 2590-c of the Education Law, as amended, L. 1970, c. 3, set out in the appendix to this opinion. This statute provides for non-partisan elections. Voters are instructed to mark next to the name of each candidate their comparative preference. Thus, assuming a field of twenty candidates, each voter may mark next to the name of a candidate any number from 1 to 20. The number 1 represents the elector's first choice, the number 2 his second, and so forth. Elections are to be conducted at a number of polling places, generally corresponding to the City's regular election districts. After the election is completed, the ballots which have been cast in each polling place are to be transported to a central counting office within the school district. Upon arrival, ballots are sorted by polling places in an order to be determined by lot. The first choices marked on the ballots are then counted, beginning with the ballots from the polling place selected by lot to be first.
*99 After all of the first preferences are counted, and the total number of valid ballots cast in the school district becomes known, a formula is applied to determine whether any candidates have been elected. To win, a candidate's votes must surpass a quota determined by dividing the total number of valid ballots cast in the school district by one more than the number of offices to be filled (in this case, the number is 10, since 9 board members are to be elected in each school district), and then adding one to this total.
If any candidate reaches this quota figure, all ballots cast for him in excess of that number are to be awarded to the second choice marked on them. The excess ballots to be transferred are those which were last credited to the candidate. Thus, ballots cast for a widely supported first choice candidate are divided into two groups upon the election of that first choice: 1) those in the polling places chosen by lot to be counted first, and 2) those in the polling places chosen by lot to be counted last. The grouping  i. e., the identity of the specific polling places to be included in each group  depends upon the number of votes needed to elect compared to the total votes counted for the candidate. Those in the first group will not be utilized in further counting; those in the second group will be counted by having second preference votes credited to a candidate, since their first choice will have already received his quota.
A similar procedure is utilized to remove candidates. First, after the transfer of surplus votes described above, all first preference votes credited to candidates receiving the fewest votes are to be transferred to the second choices indicated on the ballots. Upon completion of this transfer, the candidate remaining who has the lowest number of votes is declared defeated, and his second preference votes are transferred to the remaining undefeated candidates. Once transfers are sufficient to enable a candidate to pass the quota, he is declared elected, and all subsequent preferences attributed to him are transferred to the candidate with the next highest preference. This process is continued until all but the desired number of elected officials have been eliminated. Here, again, votes from the last polling place are transferred first. Thus, ballots are also divided into two classes by this process: 1) those cast in polling places chosen by lot to be among those counted last; and 2) those cast in polling places among those chosen by lot to be first counted. A ballot in the former group will be transferred until at least one of the choices it indicates is selected; ballots in the latter group may, in some cases, never be counted for a successful candidate.
Under transfers from both winning and losing candidates, ballots in a polling place chosen by lot to be among the last counted will have a statistically better chance that a larger number of the preferred candidates they indicate will be elected than those from a polling place chosen by lot to be among the first counted. Successive transfers merely perpetuate or increase this likelihood.
An extreme hypothetical will illustrate the plaintiffs' contention about the way in which the element of chance may effect the outcome of an election. Assume a community board district with 175 election districts, each exactly equal in population; all ballots in all districts indicate first preferences for candidate A; in 18 districts second preferences are indicated for candidate B by all voters; candidate B receives votes only in these 18 districts. In the tally of votes following the election, candidate A will receive all of the first preference votes cast in the first 18 districts to be counted (chosen randomly from the entire community district). If one or more of the 18 districts described above is among the first 17 to be counted, candidate B will be defeated. This is so because none of the second preference votes of the first 17 districts will be transferred, all being counted at first *100 preference votes for candidate A. (The votes of 17.5 districts, plus one vote, would be necessary for election.) If, however, none of the 18 districts in question are included in the first 17 to be counted, candidate B will be elected, because all of the second preference votes for him would be transferred, the first preference votes for candidate A being in excess of candidate A's quota. In considering this hypothetical it must be borne in mind that the method of selection of polling places (over 5000 spread over thirty school districts) for order of counting, as well as variations in voter preferences, would make it improbable that the hypothetical case would represent any real situation. Illustrations of how similar systems work may be found in D. Black, The Theory of Committees and Elections 72-74 (1963); C. Hoag & G. Hallett, Proportional Representation 77-89 (1926); E. Lakeman & J. Lambert, Voting in Democracies 100-12 (1955).

III. PROPORTIONAL REPRESENTATION
The method of proportional representation adopted by the State of New York in this instance is the simplest form of the Hare system of proportional representation first proposed more than a century ago. See G. Hallett, Proportional Representation  The Key to Democracy 37-57 (2d ed. 1940). The major claim made by supporters of proportional representation is that it insures that minority groups with more than negligible numbers will secure representation on any elected body. Id. at 60-61; E. Lakeman & J. Lambert, Voting in Democracies 120-22 (1955). That proportional representation may well serve this purpose is evidenced by the experience of those cities  including New York  which have at one time or another used the plan. Hallett, supra at 105-57; R. Straetz, PR Politics in Cincinnati (1958). It is almost a truism that by broadening the representative base of local school boards, community access to these elected officials will be increased. Cf. J. C. Davies, Neighborhood Groups and Urban Renewal 192 (1966).
One of the chief objections that has been advanced against proportional representation is the delay and expense involved in counting votes. See, e. g., Straetz, supra, at 20; J. Common, Proportional Representation 104 (2d ed. 1967). Some ballots will be counted and recounted in the process of elimination of candidates until they come to rest with one candidate to whose quota they have contributed.
For example, in the November, 1953 councilmanic elections in Cincinnati, Ohio, there were 19 candidates whose names were printed on the ballot. A total of 152,766 votes were cast, and thirteen counts were needed to determine the eventual nine victors. In New York City, the first election under proportional representation in 1937 resulted in an increased cost of more than $700,000. In the 1939 election, this was reduced to approximately $250,000. The time spent in counting ballots in 1937 was three weeks, while in 1939, the count took nine days. G. Hallett, Proportional Representation  The Key to Democracy 91-94 (2d ed. 1940).
Plaintiffs propose an alternate system utilizing computer technology which, they assert, would continue the benefits of proportional representation while doing away with the division of ballots into two classes based on geography. It is suggested that once a candidate achieves a quota, all of the votes which he has received should be separated according to second preferences, and those votes which exceed the quota should be distributed in proportion to the numbers of second preference votes. Such a procedure would complicate the process of counting votes. Under this system, not only would transfer ballots be counted a second time, but also non-transfer ballots. In the course of the numerous countings that proportional representation necessarily entails, this would result in the mechanical recount of many thousands of ballots in each election district. To ensure a distribution of votes that is *101 exactly proportional to voter attitudes, third, fourth, and other subsequent preferences would also have to be taken into consideration at the time of the first transfer. This is so because of the fact that the first transfer of second preference votes need not be the only transfer which these votes are to undergo. Later transfers would occur as candidates to whom votes have been transferred were eliminated. The complexity of this task is a factor that legislators and their staffs might well take into account.
While computers probably could handle the problem with facility, their use would, legislators might find, have some disadvantages. However accurate they might be, some candidates and some voters might suspect tampering with programming. Voters and candidates conceivably could prefer a less efficient system which could be visually observed by the press and public and which would result in payment of wages to locally hired watchers and counters rather than to stranger computer technicians. It is significant that the right of the public to attendance at the count is explicitly provided. Paragraph 31, subsection 6, L. 1969, c. 330, § 4.
Another alternative referred to with favor by the plaintiffs at the hearing would achieve approximately the same statistical result without necessitating so prolonged or complex a recount process. Under this system, employed in Cincinnati for several years, the total number of first preference votes received by a candidate is divided by the number of the votes which he received in excess of his quota. The resulting figure, rounded to the next-highest integer is used to determine which ballots are to be transferred. Of the total number of ballots, every one which corresponds to this resultant integer is assigned to the transfer. Thus, if the integer is 3, every third ballot is transferred. The process continues through the total number of ballots received by the transferor candidate, and is repeated, with a new integer arrived at by dividing the total number of ballots by the number of remaining votes to be transferred, until the requisite number have been reassigned. This assures that all geographic areas are represented because the transferred ballots are selected from all of the ballots cast at all of the polling places. The process is more time-consuming and expensive than that selected by New York.
Random selection of polling places has been utilized in New York City before. When City councilmanic elections were held on a borough-wide basis according to the system of proportional representation, the ordinal distribution of polling places was quite even over each of the boroughs. Polling places from widely scattered areas were to be found among those first chosen, as well as among those last chosen. In itself, the random nature of the selection process tends to ensure that the ballots which are counted are representative of all ballots cast. See G. Hallett, Proportional Representation  The Key to Democracy 99 (1940). Cf. Johnson v. City of New York, 274 N.Y. 411, 9 N.E.2d 30, 35 (1937) (the question of unconstitutionality because of chance in counting raised in the briefs before the Court of Appeals (Brief for Democratic County Committee 22), was not specifically discussed in its opinion).
Testimony indicates that in Cincinnati before adoption of the procedure discussed above, counting proceeded on a purely geographical basis, with the same wards counted first every time. No significant distortion of results was noted; the procedures were changed, one of defendants' witnesses testified, to avoid the possibility of unsubstantiated charges of distortion.
In addition to the methods already described witnesses referred with more or less approval to variations used in such diverse locations as Boulder, Worcester, Tasmania and Ireland as well as to a number of untried schemes. Obviously, the New York Legislature had a wide field from which to choose.
As witnesses for plaintiffs conceded, the element of chance is present, to a *102 greater or lesser degree, in all election systems. Thus, even in a single-member constituency  the ordinary system used in the United States today  the drawing of district lines can result in shifting a voter from membership in a majority whose vote elects his choice, to membership in a minority whose candidate is defeated. If the changing of district lines to reflect changing populations is not deliberately engineered to take advantage of voting patterns, but is purely arbitrary, the creation of a majority may well result from chance.
Two of the plaintiffs' witnesses were experts in mathematics and probability though they had made no study of proportional representation before being retained in this litigation. They testified persuasively that models might be developed to predict the effect of variations in various parameters and that changes in some of the parameters might affect the results of an election. The third witness for plaintiffs testified that patterns of candidates and groupings of candidates were just beginning to develop in the area where her children attended school. But none of the plaintiffs' experts were able to apply their mathematical analysis to objective data about candidates, voters, sociology, geography, or anything else. The "intuitive feeling"  as he himself phrased it  of a mathematician witness that the order of counting would probably affect the election hardly suffices as a factual predicate for a declaration of unconstitutionality.
By contrast, the two expert witnesses for the defendants had had extensive experience with counting ballots in proportional representation elections in a number of cities; they had both published studies of such systems. Their opinion was that order of counting would have no appreciable affect on who was elected.
Experts for both sides agreed that all election schemes which have been devised share the common failing that none can be said with certainty to always and inevitably reflect the "will" of the electors. For example, the Borda system of balloting urged by one of plaintiffs' witnesses operates upon the assumption that preferences among candidates are quantifiable and uniform. But, it is highly doubtful that electors "mark" candidates, and, if they do, it is even more unlikely that they "mark" them uniformly. Cf. D. Black, The Theory of Committees and Elections 65 (1963). Yet, the Borda system accords uniform weight to each preference. Further, there is no way to supervise a voter to ensure that he indicates a comparative preference for all candidates. The temptation, under the Borda system, would be to vote only for a first preference candidate, knowing that any indication of a second or other preference would result in a benefit to these candidates, to the possible detriment of the first preference candidate.

IV. EQUAL PROTECTION
In determining the applicability of the Equal Protection Clause of the Fourteenth Amendment two related tests may be utilized. First, a state law is unconstitutional if it invidiously discriminates against a class of citizens. Harper v. Virginia State Board of Elections, 383 U.S. 663, 666, 86 S.Ct. 1079, 16 L.Ed.2d 169 (1966); Carrington v. Rash, 380 U.S. 89, 85 S.Ct. 775, 13 L. Ed.2d 675 (1965). Second, a discrimination may be justified by some compelling state interest, if it is rational and not arbitrary. Kramer v. Union Free School District No. 15, 395 U.S. 621, 626-627, 89 S.Ct. 1886, 23 L.Ed.2d 583 (1969); Walters v. City of St. Louis, 347 U.S. 231, 74 S.Ct. 505, 98 L.Ed. 660 (1954).
In some instances, state discriminations may be inherently invidious. See, e. g., Brown v. Board of Education, 347 U.S. 483, 74 S.Ct. 686, 98 L.Ed. 873 (1954). Where fundamental rights and liberties are threatened, states may be required "to tailor carefully the means of satisfying a legitimate state interest." Katzenbach v. Morgan, 384 U.S. 641, 654, n. 15, 86 S.Ct. 1717, 1726, 16 L.Ed.2d 828 (1966). Among these *103 basic and fundamental rights is the right to vote. Reynolds v. Sims, 377 U. S. 533, 562, 84 S.Ct. 1362, 12 L.Ed.2d 506 (1964); Moore v. Ogilvie, 394 U.S. 814, 89 S.Ct. 1493, 23 L.Ed.2d 1 (1969). While some limitations upon the right to vote may be permissible (see, e. g., Lassiter v. Northampton County Board of Elections, 360 U.S. 45, 79 S.Ct. 985, 3 L.Ed.2d 1072 (1959) (literacy test); Gray v. Sanders, 372 U.S. 368, 380-81, 83 S.Ct. 801, 9 L.Ed.2d 821 (1963) (dictum; "minors, felons, and other classes may be excluded")), once a state chooses to extend this right, all are entitled to participate equally. Harper v. Virginia State Board of Education, 383 U.S. 663, 665, 86 S.Ct. 1079, 16 L.Ed.2d 169 (1966); Avery v. Midland County, Texas, 390 U.S. 474, 480, 88 S.Ct. 1114, 20 L.Ed.2d 45 (1968). That these rules apply to local school board elections is clear. Kramer v. Union Free School District No. 15, 395 U.S. 621, 89 S.Ct. 1886, 23 L.Ed.2d 583 (1969); Oliver v. Board of Education, 306 F.Supp. 1286 (S.D.N.Y.1969). These general rules apply to discrimination among candidates as well as among voters. Williams v. Rhodes, 393 U.S. 23, 89 S.Ct. 5, 21 L.Ed.2d 24 (1968); Moore v. Ogilvie, 394 U.S. 814, 89 S.Ct. 1493, 23 L.Ed.2d 1 (1969). Thus, the questions which must be answered are two: does any discrimination within the meaning accorded that word under the Equal Protection Clause exist in the statutory scheme adopted; if such a discrimination exists, is it invidious? Cf. McDonald v. Board of Election Commissioners of Chicago, 394 U.S. 802, 89 S.Ct. 1404, 22 L.Ed.2d 739 (1969).
In all of the cases applying the Equal Protection Clause to invalidate state election laws, there is none that fails to contain an identifiable group which is the subject of the alleged discrimination. See, e. g., Carrington v. Rash, 380 U.S. 89, 85 S.Ct. 775, 13 L.Ed.2d 675 (1965) (servicemen); Harper v. Virginia State Board of Elections, 383 U.S. 663, 86 S. Ct. 1079, 16 L.Ed.2d 169 (1966) (those financially unable to pay a poll tax); Reynolds v. Sims, 377 U.S. 533, 84 S.Ct. 1362, 12 L.Ed.2d 506 (1964) (those living in heavily populated districts); Kramer v. Union Free School District No. 15, 395 U.S. 621, 89 S.Ct. 1886, 23 L.Ed.2d 583 (1969) (those who did not own property). The repeated use of the word "invidious" in these cases is not without significance. But cf. Michelman, On Protecting the Poor Through the Fourteenth Amendment, 83 Harv.L. Rev. 7, 26-33 (1969). In each of its meanings, "invidious"  as its root of envy suggests, and as its connotations of jealousy, hate and defamation confirm  conveys the idea of being directed against some persons or groups. And, of course, "discrimination" implies an object discriminated against.
Under the statute now being challenged there is no identifiable group being discriminated against. At the time that ballots are cast, the order of polling places remains unknown. At that time, each individual has exactly the same statistical probability of being included in an early-count polling place. Each stands an equal opportunity of being benefitted or injured by the lottery. No identifiable individual or interest group is disenfranchised. The large number of polling places within each school board district insures that group interests will be adequately represented both at the top and bottom of the lists.
The fact that an element of chance is incorporated into a method of election is not necessarily violative of the Equal Protection Clause. See, e. g., Moore v. Election Commissioners of Cambridge, 309 Mass. 303, 35 N.E.2d 222 (1941); Johnson v. City of New York, 274 N.Y. 411, 9 N.E.2d 30 (1937). As noted above, chance is an element of most election procedures. The Archons of ancient Athens would undoubtedly have been surprised to learn that their selection by lot from among the citizenry was the result of an invidious discrimination. See, 2 Ency. Britannica, Archon 329 (1967). The various United States District Courts which have adopted random *104 jury selection plans pursuant to Congressional mandate might be disturbed to find that in their efforts to secure to all an equal opportunity for jury service they have actually been violating the concept of equal protection. See 28 U.S.C. § 1863. The new random selection system for selective service registrants is but another example of a process in which chance is utilized to ensure an equal distribution of rights and duties. See 83 Stat. 220 (1969).
In short, in the circumstances of this case, there can be no discrimination when there is no one discriminated against. There can be no denial of equal protection when all share an equal opportunity to have their votes count in an election.
Even if it were to be held, arguendo, that there does exist a discrimination against those who are eventually chosen to be counted first, it cannot be maintained that such a discrimination is invidious. The state plan that has been adopted is reasonable and is designed to reach the goals that are embodied in the entire statutory scheme of decentralization. As noted above, the purpose of decentralization in New York City was to secure widespread community representation and participation in the governance of the City schools. Since school elections are to be conducted at times different from those at which the regular elections are held, it is reasonable to assume that many voters, some of whom are intended as the prime beneficiaries of decentralization, will not participate. Proportional representation assures that these people will secure at least minimal representation on the local boards, since it is expressly designed for the preservation of a minority voice in government. In re Ross, 175 Misc. 43, 21 N.Y.S.2d 848, 850 (Sup.Ct.1940).
The New York Legislature may reasonably conclude that proportional representation will help secure as widespread support for the decisions reached by the school boards as possible. It was not irrational for it to believe that each individual, knowing that a representative for whom he has indicated a preference will participate in decision making, is more likely to acquiesce in the result. The legislature had adequate grounds to find that proportional representation is better adapted to carry out the purposes of decentralization than majority or plurality elections.
If  as we continue to assume for purposes of this argument  distinctions are built into the system, the state must insure that unnecessarily harsh discriminations do not result from the method chosen to achieve a proper goal. See Shelton v. Tucker, 364 U.S. 479, 488, 81 S.Ct. 247, 5 L.Ed.2d 231 (1964); cf. Elfbrandt v. Russell, 384 U.S. 11, 18, 86 S.Ct. 1238, 16 L.Ed.2d 321 (1966). See generally Note, Less Drastic Means and the First Amendment, 78 Yale L.J. 464 (1969); Struve, The Less Restrictive Alternative and Economic Due Process, 80 Harv.L.Rev. 1463 (1967). But, an incidental and minor discrimination is justified in the absence of any appreciably more suitable alternate plan. As the Court of Appeals for this circuit recently remarked in a voting case,
Even if we * * * assume that the alternative means of voting [outside the regular polling place in a church] places a burden on the free exercise * * * of * * * religion, we would nevertheless have to conclude that any incidental burden is so slight that it does not begin to outweigh the interest of the state in having available to it the additional polling places which the use of the churches affords. Berman v. Board of Elections, 420 F.2d 684 (2d Cir. 1969).
The interest that must be served in any election system is the desire to put to rest disputes and differences, both within the community and within the City as a whole. If the procedure suggested by the plaintiffs, or the procedure utilized in Cincinnati were used, the inevitable delay might, in the legislature's reasonable view, cause these differences to continue to smolder. Community *105 tensions might increase and community wounds fester with substantial delays in vote counts. Additionally, the inevitable errors, much less the possibility of intentional interference, involved in the constant manipulation and recounting of ballots that would be necessitated by some of the alternate systems suggested could intensify suspicions and hostilities.
Finally, the proposed alternatives are unlikely to achieve any more accurate a reflection of voter desires in the community districts than the procedure presently under attack. It is highly improbable that the plan advanced by the plaintiffs will serve to achieve more than a marginal, de minimis, improvement.
None of the witnesses for the plaintiffs would hazard an opinion upon the degree of probability that any candidate's election would be affected by choice of a system for counting ballots. All that they could do was suppose that if the election in a single school district had a certain percentage of probability of being affected by order of count, then the probability of the citywide election for all thirty boards being affected was another percentage. The utilization of hypothetical and unverified probabilities makes suspect conclusions reached even by impeccable mathematical manipulation since the conclusions are no more reliable than the assumptions. See, e. g., People v. Collins, 68 Cal.2d 319, 66 Cal.Rptr. 497, 438 P.2d 33 (1968) (en banc); Miller v. State, 240 Ark. 340, 399 S.W.2d 268 (1966). See also, Maxwell v. Bishop, 398 F.2d 138, 141-48 (8th Cir.), cert. granted, 393 U.S. 997, 89 S.Ct. 488, 21 L.Ed.2d 462 (1968). But cf. Finkelstein and Fairley, A Bayesian Approach to Identification Evidence, 83 Harv.L.Rev. 489, 516-17 (1970). The computations offered at the trial were  unlike some of those by Professor Banzhaf relied upon by the New York Court of Appeals in Iannucci v. Board of Supervisors, 20 N.Y.2d 244, 251, 282 N.Y.S.2d 502, 507, 229 N.E.2d 195, 198 (1967)  not based upon any model demonstrably reflective of a real situation. See, e. g., Banzhaf, Weighted Voting Doesn't Work: A Mathematical Analysis, 19 Rutgers L.Rev. 317 (1964).
That any substantial voter preference for second choice candidates will be confined solely or primarily to the first polling places chosen by lot would be highly unlikely. The procedure adopted by the state assures that a fairly accurate reflection of voter preferences will be achieved, since random selection of the large number of polling places secures a fair representation of second and subsequent preferences as well as of first place preferences.

V. DUE PROCESS
The attack under the Due Process Clause of the Fourteenth Amendment must also fail. Accepting plaintiffs' contention that the right to the franchise is one of the many unenumerated rights encompassed in the term "liberty" that is protected by due process (cf. Shapiro v. Thompson, 394 U.S. 618, 89 S.Ct. 1322, 22 L.Ed.2d 600 (1969); Griswold v. Connecticut, 381 U.S. 479, 85 S.Ct. 1678, 14 L.Ed.2d 510 (1965); United States v. Tarlowski, 305 F.Supp. 112 (E.D.N.Y.1969)), no arbitrary or otherwise constitutionally impermissible limitation has been placed upon that right by the New York statute. Much of what has been said in regard to the rationality of the counting procedure under the Equal Protection Clause supports this conclusion. Chance, if a rational basis exists for its employment, cannot be said to be an irrational factor. At most, it is non-rational or a-rational, taking on its coloration from the purpose to which it is put. Here, that purpose is a rational one. No deprivation of the fundamental "liberty" of the ballot has been accomplished without due process.

VI. CONCLUSION
It should be borne in mind that the program of decentralization that has so recently been adopted for New York schools is untried and untested. There *106 is need for experiment and innovation to meet the critical urban and educational problems of our society. Cf. Avery v. Midland County, Texas, 390 U.S. 474, 485, 88 S.Ct. 1114, 20 L.Ed.2d 45 (1968). The proportional representation system chosen by New York State in this instance is but one of the manifestations of the diversity and initiative encouraged by our Constitution and inherent in our federal system. Too rigid and too quick an exercise of judicial supervision through application of mechanical rules of constitutional construction may have an unfortunate inhibiting effect on legislators charged in the first instance with responsibility for meeting new challenges. So long as no constitutional right of any individual or group is adversely affected to any discernable degree, the states' rights to experiment under the Tenth Amendment may not be limited by federal courts.
Plaintiffs have failed to show a reasonable probability that they can prove a cause of action upon which relief may be granted. Accordingly, their motion for a preliminary injunction is denied.
So ordered.

APPENDIX
L.1969, c. 330, § 4, as amended, L.1969, c. 422, § 1, as amended, L.1970, c. 3, § 5:

*   *   *   *   *   *
6. The members of each community board shall be elected by proportional representation in accordance with the following rules:
(1) Nomination by petition. Candidates for community board member shall be nominated by petitions in accordance with regulations, promulgated by the city board, and approved by the board of elections in the city of New York. Such petitions shall be filed with the board of elections at least four weeks before the election.
(2) Number of signatures. Such nominating petition shall be signed by not fewer than two hundred registered voters residing in such community district, or persons eligible to vote as parents in such community district pursuant to subdivision three of this section.
(3) Separate petitions and signers. Each candidate shall be nominated by a separate petition and no elector shall sign more than one such petition. Should an elector sign more than one such petition, his signature shall be void except upon the petition first signed.
(4) Nonpartisan petitions. No candidate shall be identified by political party or other organizational affiliation on the nominating petitions.
(5) Paper ballots. Community board members shall be voted for, . . . on paper ballots on which the candidates are listed by name only. The ballots shall conform to the provisions of the election law for paper ballots, so far as applicable, * * *
The ballots shall contain a square for voting before each candidate's name.
(6) Order of names on ballot. The names of the candidates shall be printed in the alphabetical order of their surnames, except that they shall be rotated by polling places by transposing the first named candidate to the bottom of the order at each succeeding polling place; so that each name shall appear first and in each other position in an equal number, as nearly as possible, of the polling places.

*   *   *   *   *   *
(8) Central count. Prior to every election at which community board members are to be elected, the board of elections shall designate a central counting place for each community district where the ballots shall be brought together and counted publicly; shall appoint for each district a board of two competent persons, to act as directors of the count for such district; shall employ a sufficient staff of assistants for each district, and shall make suitable arrangements for the counting and recording of the ballots, subject to the provisions of this article. The board of elections shall prepare and provide all necessary forms and equipment.
*107 (9) Assembling ballots. As soon as the polls have closed, the election officials assigned by the board of elections at each polling place shall seal the ballot boxes without opening them and shall send them at once, as the board of elections may direct, to the central counting place for the district with a record of the number of ballots for community board member which have been voted in their polling place.
(10) Checking number of ballots. At the central counting place the number of ballots for community board member found in each ballot box shall be recorded and compared with the record sent from the corresponding polling place. The records thus compared shall be made available to the public with notations explaining any corrections or changes made therein. Discrepancies which cannot be reconciled shall be shown on the record. All ballots found in the ballot boxes which bear no evidence of having been improperly cast shall be accepted.
(11) Sorting of ballots. Ballots shall be sorted by polling places in an order determined by lot.
(12) Rules for validity. If a ballot does not clearly show which candidate the voter prefers to all others or if it contains the signature of the voter, it shall be held as invalid. Every ballot not thus invalid shall be counted according to the intent of the voter so far as that can be clearly ascertained, whether marked according to the instructions printed on it or not. No ballot shall be held invalid because it is marked in ink or pencil different from the one supplied at the polling place, or because the names of candidates thereon have been stricken out by the voter. Any cross mark or check mark shall be disregarded, except that a single cross mark or check mark on a ballot on which no number one appears shall be considered equivalent to the number one. If the consecutive numerical order of the numbers on a ballot is broken by the omission of one or more numbers, the smallest number marked shall be taken to indicate the voter's first choice, the next smallest his second, and so on, without regard to the number or numbers omitted.
(13) Count of first choices. At the beginning of the count for each district the ballots shall be sorted and counted according to the first choices marked on them. The ballots shall be so credited to the candidates of their choice in the order of polling places chosen by lot as specified in paragraph eleven of this subdivision. The number of valid ballots cast for each candidate as first choice in each polling place and the total number of valid ballots for each candidate and for all candidates shall be determined and recorded.
(14) Single transferable vote. Each candidate shall be credited with one vote for every ballot that is sorted to him as first choice or transferred to him as hereinafter provided, and no ballot shall ever be credited to more than one candidate at the same time.
(15) Quota sufficient to elect. The quota of votes sufficient to elect a community board member shall be determined by dividing the total number of valid ballots cast in the community district by one more than the number of members to be elected for the district and adding one to the result, disregarding fractions. This is the smallest number of ballots which could be received separately by each of as many candidates as are to be elected but not by one more.
(16) Election of candidates with quotas. All candidates whose first-choice ballots equal or exceed the quota shall be declared elected.
(17) Transfer of surplus ballots. All of the surplus ballots in excess of the quota of each candidate so elected shall be transferred from him each to the unelected candidate indicated on it as next choice among such candidates. The ballots to be so transferred as surplus ballots shall be those last received by the candidate in the count of first choices which show a clear next choice for an unelected candidate. All ballots which show no such clear next choice shall be left to the credit of the candidate of their first choice. If more than one *108 candidate has first-choice ballots in excess of the quota, the surplus ballots of the candidate with most ballots shall be transferred first, then those of the candidate with next most ballots, and so on.
(18) Election of candidates during transfers. Whenever during any transfer of ballots, at any stage of the counting, the number of ballots credited to a candidate becomes equal to the quota, he shall be declared elected and no ballots in excess of the quota shall be transferred to him. Any transferred ballots in excess of the quota which show a next choice for such candidate shall be transferred further at once, each to the next subsequent choice on it for a continuing candidate. A "continuing candidate" is a candidate not yet elected or defeated. If such a ballot shows no such further choice, it shall be set aside as "exhausted".
(19) Defeat of lowest candidates. After the count of first choices and the transfer of all surplus ballots, if any, the candidate having the fewest votes to their credit shall be successively defeated and their ballots transferred as hereinafter provided. The one candidate with the fewest votes shall be declared defeated first. If at this point, two or more of the candidates with the next fewest votes, including any such candidates whose names have been written in, have together fewer votes than the candidate next higher in number of votes, they may all be declared defeated together unless this would reduce the number of undefeated candidates below the number to be elected.
(20) Transfer of ballots from defeated candidates. All the ballots of the candidates thus defeated shall be transferred, each to the candidate indicated on it as next choice among the continuing candidates. If a ballot shows no such further choice, it shall be set aside as exhausted. If the same choice is marked for more than one candidate, it shall be disregarded except as to continuing candidates, but if the next choice for a continuing candidate is marked for more than one continuing candidate, the ballots shall be set aside as exhausted.
(21) Defeat of candidate then lowest. When all the ballots of the candidate or candidates first defeated have been transferred, the one candidate who is then lowest on the poll shall be declared defeated and all his ballots transferred in the same way.
(22) Successive defeats and transfers of ballots. Thereupon the candidate who is then lowest on the poll shall be declared defeated and all his ballots similarly transferred. The lowest candidates shall be declared defeated one at a time and all their ballots transferred until the election is at an end as hereinafter provided.
(23) Order of transfer. When ballots are being transferred from defeated candidates, they shall be transferred in the reverse order to that in which they were credited to the candidate whose ballots are being transferred, except that if no quota can possibly be completed for another candidate during the transfer they may be transferred in any order.
(24) Ties. In deciding any tie a candidate shall be treated as having more votes than another if he was credited with more votes at the end of the last preceding transfer or sorting of ballots at which the numbers of their votes were different. Any tie not thus decided shall be decided by lot.
(25) Election ended when all quotas are completed. If at any time as many candidates as are to be elected have received the quota, the other candidates shall all be declared defeated and the election shall be at an end. Any transfer that is in progress when the last candidate is elected may be completed for the record.
(26) Last candidates elected even if quotas are not completed. If at any time all ballots of any defeated candidates have been transferred and it is impossible to defeat another candidate without reducing the continuing candidates below the number still to be elected, all the continuing *109 candidates shall be declared elected and the election shall be at an end.
(27) Correction of errors. If at any time after the first sorting of the ballots a ballot is found to have been misplaced, it shall be credited to the candidate who should have been credited with it at that stage of the counting or set aside as exhausted if that would have been the proper disposition of it at that stage, and any changes in the disposition of the ballots composing completed quotas made necessary by the correction shall also be made forthwith. If the number of misplaced ballots found indicates that the list of continuing candidates may be incorrect, so much of the sorting and counting as may be required to correct the error shall be done over again before the count proceeds.
(28) Record of count. A record of the count shall be kept in such form as to show, after each sorting or transfer of ballots, the number thereby credited to each candidate, the number thereby found exhausted, the total for each candidate, the total found exhausted, and the total number of valid ballots found by adding the totals of all candidates and the total found exhausted.
(29) Record and disposition of ballots. Every ballot that is transferred from one candidate to another shall be stamped or marked so as to show all the candidates to whom it is successively credited during the entire course of the count. If in correcting an error, or in recounting ballots, any ballots are resorted or re-transferred, every such ballot shall be made to take the same course that it took in the original count unless the correction of an error requires its taking a different course.

*   *   *   *   *   *
(32) Supplementary regulations. Administrative regulations for the conduct of elections by proportional representation, not inconsistent with the provisions of this article, may be made by the city board and, subject to any such regulation, by the board of elections.
(33) Rights of candidates. At each election any candidate for community board member shall be entitled, upon written application to the board of elections at least five days before said election:
(a) To exercise all the rights granted by the election law to a political party in regard to the appointment of watchers and challengers for the polls. Such watchers and challengers may exercise their respective rights at the polls until the ballots have been sent to the central counting place and may accompany the ballot boxes to the central counting place.
(b) To appoint two representatives at the count in the central counting place, who shall have full authority to move anywhere within the central counting quarters for the district, to inspect all activities of the count without interfering therewith and to exercise all rights conferred on watchers under the election law.
(c) To appoint two observers at the central counting place, who shall be given facilities for keeping in full view all ballots outside of containers and all containers of ballots at all times when such ballots are not being sorted or counted, from the time when the first ballots arrive until all ballots have been placed in containers and removed for safe-keeping at the end of the count.
The board of elections shall permit substitutions for persons originally appointed.
(34) Public attendance at count. The candidates, representatives of the press and other media and, so far as may be consistent with good order and convenience, the public shall be afforded every facility for being present and witnessing the count.

*   *   *   *   *   *
