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 Voting power distribution (Posted on 2005-01-24)
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?

 See The Solution Submitted by Tristan Rating: 3.7143 (7 votes)

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 re: computer solution | Comment 5 of 13 |
(In reply to computer solution by Charlie)

A modification of the program shows the actual voting patterns that will win for each of the weight distributions.  In the case of distributions allowing ties, another line shows the voting patterns that lead to ties.  In each case, it is assumed that the person with the most votes (voting weight) votes Y.  The patterns shown are the patterns necessary to tilt the final result to Y, or to tie on the case of the second line for that type of distribution.

`  1  1  1  1  1      5    3      1NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY   2  1  1  1  1      6    3                  5NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNNY NNYN NYNN YNNN   2  2  1  1  1      7    4     10NNYY NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY   3  1  1  1  1      7    4      5NNNY NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY   2  2  2  1  1      8    4                 10NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNYY NYNN YNNN   3  2  1  1  1      8    4                 20NNYY NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNNY NNYN NYNN   4  1  1  1  1      8    4                  5NNNY NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNNN   3  2  2  1  1      9    5     30NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY   5  1  1  1  1      9    5      5NNNN NNNY NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY   3  2  2  2  1     10    5                 20NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNYN NYNN YNNN   3  3  2  1  1     10    5                 30NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNYY NYNN   4  2  2  1  1     10    5                 30NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNNY NNYN   3  3  3  1  1     11    6     10NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY   4  2  2  2  1     11    6     20NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY   3  3  2  2  2     12    6                 10NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY YNNN   4  3  2  2  1     12    6                 60NNYY NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNYN NYNN   4  3  3  1  1     12    6                 30NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNYY   5  2  2  2  1     12    6                 20NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNNY   4  3  3  2  2     14    7                 30NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY NYNN YNNN   5  3  3  2  1     14    7                 60NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NNYN   5  4  3  2  2     16    8                 60NNYY NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY NYNN                                 81         390`

 Posted by Charlie on 2005-01-25 01:49:14
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