The five owners of Plexus and Co. are voting on a very important decision (Top secret!). Each must vote for or against the decision. They don't necessarily own equal shares of the company, so they don't necessarily have equal voting power. For example, one person might have 5 votes and the other four have 1 vote each. However, it is distributed in a way that a tie is impossible. Obviously, everyone has positive voting power.

There are 2^5=32 different ways that the five people can vote (such as YYNNY, YNNYY, NNNNN, ...). Each way will result in favor or against the decision, depending on how the voting power is distributed.

There are 2^32 different combinations of the 32 outcomes, but not every combination is possible. For example, it is impossible for YYYNN to be in favor of the decision while YYYYN is against the decision, no matter how the voting power is distributed.

Out of the 2^32 different combinations, how many are possible, remembering that combinations where a tie is possible are not allowed?

The resulting votes,  YYNNN - YNYNN - YNNYY - YNNYN - YNNNY - YNNNN
encompass the different voting outcomes that can occur with each shareholder voting (each having no more than five shares or fractional shares), with no possibility of a tie, in a Plexus and Co. meeting. The vote arrangements that are not included are those that are mirror images of those that are (e.g., NYYYY is the mirror of YNNNN where a "Yea" outcome of one will be a "Nay" outcome of the other) and those that do not rely on or seems unaffected by the distribution (e.g., YYYYY will always result in a "Yea" outcome).  

There are 7 different voting distribution outcomes  for the above resulting votes, they are:
     (a) N-N-Y-N-N-N
   (b) Y-N-Y-N-N-N
   (c) Y-Y-N-N-N-N
   (d) Y-Y-Y-N-N-N
   (e) Y-Y-Y-Y-N-N
   (f) Y-Y-Y-Y-Y-N
   (g) Y-Y-Y-Y-Y-Y

The possible voting distributions and those that can result in a tie follows:
11111    1    a
21111       5 -
22111   10    b
22211      10 -
22221    5    a
22222    1    a
31111    5    f
32111      20 -
32211   30    d
32221      20 -
32222    5    a
33111   10    b
33211      30 -
33221   30    b
33222      10 -
33311   10    c
33321      20 -
33322   10    a
33331    5    a
33332    5    a
33333    1    a
41111       5 -
42111   20    f
42211      30 -
42221   20    e
42222       5 -
43111      20 -
43211   60    d
43221      60 -
43222   20    b
43311      30 -
43321   60    d
43322      30 -
43331      20 -
43332   20    a
43333    5    a
44111   10    b
44211      30 -
44221   30    b
44222   10    b
44311   30    c
44321      60 -
44322   30    b
44331   30    b
44332      30 -
44333   10    a
44411   10    c
44421   20    c
44422      10 -
44431      20 -
44432   20    a
44433   10    a
44441    5    a
44442    5    a
44443    5    a
44444    1    a
51111    5    g
52111      20 -
52211   30    f
52221      20 -
52222    5    f
53111   20    f
53211      60 -
53221   60    e
53222      20 -
53311   30    d
53321      60 -
53322   30    d
53331   20    e
53332      20 -
53333    5    a
54111      20 -
54211   60    d
54221      60 -
54222   20    b
54311      60 -
54321  120    d
54322      60 -
54331      60 -
54332   60    b
54333      20 -
54411   30    c
54421      60 -
54422   30    d
54431   60    d
54432      60 -
54433   30    a
54441      20 -
54442   20    a
54443   20    a
54444    5    a
55111   10    b
55211      30 -
55221   30    b
55222   10    b
55311   30    c
55321      60 -
55322   30    b
55331   30    b
55332   30    b
55333   10    b
55411   30    c
55421   60    c
55422      30 -
55431      60 -
55432   60    b
55433      30 -
55441   30    b
55442      30 -
55443   30    a
55444   10    a
55511   10    c
55521   20    c
55522   10    c
55531   20    c
55532      20 -
55533   10    a
55541      20 -
55542   20    a
55543   20    a
55544   10    a
55551    5    a
55552    5    a
55553    5    a
55554    5    a
55555    1    a

The first column is the possible share distribution among the five shareholders. Those in black are the possible share distributions and those in red are those that can result in a tie. The second and third columns represent the permutations of the given vote by assigning the different voting shares to other shareholders. The second column represents those permutations where the voting outcome can not result in a tie and the third column represents those that can result in a tie. The fourth column represents the different voting outcome the voting distribution to which to share distribution is a category.

The total permutations, where the voting outcome does not result in a tie, is 1760 of the 3125 (5⁵) possible different share distributions.

Edited on November 18, 2007, 1:00 am

Posted by Dej Mar on 2007-11-17 00:40:24