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Voting power distribution (Posted on 2005-01-24) Difficulty: 4 of 5
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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Solution 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      1
NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY
  2  1  1  1  1      6    3                  5
NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNNY NNYN NYNN YNNN
  2  2  1  1  1      7    4     10
NNYY NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
  3  1  1  1  1      7    4      5
NNNY NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
  2  2  2  1  1      8    4                 10
NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNYY NYNN YNNN
  3  2  1  1  1      8    4                 20
NNYY NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNNY NNYN NYNN
  4  1  1  1  1      8    4                  5
NNNY NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNNN
  3  2  2  1  1      9    5     30
NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
  5  1  1  1  1      9    5      5
NNNN NNNY NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
  3  2  2  2  1     10    5                 20
NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNYN NYNN YNNN
  3  3  2  1  1     10    5                 30
NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNYY NYNN
  4  2  2  1  1     10    5                 30
NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNNY NNYN
  3  3  3  1  1     11    6     10
NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
  4  2  2  2  1     11    6     20
NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
  3  3  2  2  2     12    6                 10
NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY
YNNN
  4  3  2  2  1     12    6                 60
NNYY NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNYN NYNN
  4  3  3  1  1     12    6                 30
NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNYY
  5  2  2  2  1     12    6                 20
NNYN NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNNY
  4  3  3  2  2     14    7                 30
NNYY NYNY NYYN NYYY YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NYNN YNNN
  5  3  3  2  1     14    7                 60
NNYY NYNN NYNY NYYN NYYY YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
NNYN
  5  4  3  2  2     16    8                 60
NNYY 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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