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?

Some models of this can be put into a diagram. The top stem part of the junction equals what the majority of the other branches are. For example, 22111 and 11111 could look like this:

  |      |
2-|-2  1-|-1
  |     /|\
1-|-1  1 1 1
  |
  1

This means the majority of what the 3 1s think is equivalent to one 2-size vote. It will only matter if the two 2s disagree. The five 1s branch off from the same junction and so all their votes are of equal value. 33111 and 51111 can be diagrammed easily the same way. However, these don't support parts where only a majority of the contributing voters can outrule, like 31111 and 42221 and it also doesn't cover the strange distribution 32211.

Posted by Gamer on 2005-01-25 20:41:28