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?
(In reply to
Perplexed by Richard)
Let S be the set of 32 five-letter sequences of Y's and N's.
Think of a "combination" as a mapping of S into {approve, disapprove}.
For example, if person 1 has 5 votes and the others have 1 vote each, the mapping f looks like:
f(YYYYY) = approve
f(YYYYN) = approve
f(YYYNY) = approve
f(YYNYY) = approve
f(YNYYY) = approve
f(YYYNN) = approve
f(YYNYN) = approve
f(YNYYN) = approve
f(YYNNY) = approve
f(YNYNY) = approve
f(YNNYY) = approve
f(YYNNN) = approve
f(YNYNN) = approve
f(YNNYN) = approve
f(YNNNY) = approve
f(YNNNN) = approve
f(NYYYY) = disapprove
f(NYYYN) = disapprove
f(NYYNY) = disapprove
f(NYNYY) = disapprove
f(NNYYY) = disapprove
f(NYYNN) = disapprove
f(NYNYN) = disapprove
f(NNYYN) = disapprove
f(NYNNY) = disapprove
f(NNYNY) = disapprove
f(NNNYY) = disapprove
f(NYNNN) = disapprove
f(NNYNN) = disapprove
f(NNNYN) = disapprove
f(NNNNY) = disapprove
f(NNNNN) = disapprove
There are 2^32 such mappings. Your task is to determine how many of those correspond to voting power allocations. The problem statement gives one example of a mapping that is not possible.
Hope that clears things up.
Edited on January 24, 2005, 10:13 pm
Clarification
