Problem:
A permutation of is heavy-tailed if . What is the number of heavy-tailed permutations?
Answer Choices:
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Solution:
Call a permutation balanced if , and consider the number of balanced permutations. The sum of all five entries is odd, so in a balanced permutation, must be 1,3 , or 5 . For each choice of , there is a unique way to group the remaining four numbers into two sets whose elements have equal sums. For example, if , the two sets must be and . Any one of the four numbers can be , and the value of is then determined. Either of the two remaining numbers can be , and the value of is then determined. Thus there are balanced permutations of
, and permutations that are not balanced. Call a permutation heavy-headed if . Reversing the entries in a heavy-headed permutation yields a unique heavy-tailed permutation, and vice versa, so there are exactly as many heavy-headed permutations as heavy-tailed ones. Therefore the number of heavy-tailed permutations is .
The problems on this page are the property of the MAA's American Mathematics Competitions