Let's say I have objects, each with a number from 1 to 5 printed on them. There are objects with the number printed on them for . These objects are otherwise indistinguishable. Let be the set of all permutations of these objects ( is essentially a set of multiset permutations). Given any permutation , let be the number of times the number appears on any of the first objects in the permutation, and let be the number of times the number appears on any of the last objects in the permutation. Fix an integer . Let .
Evaluate the following sum:
I already solved this problem, but it involved summing over all integral solutions to a Diophantine equation in 5 variables. I am looking for an alternate solution. Here is my attempt to solve it (any advice would be greatly appreciated). Since each number must appear times in any permutation, . Hence, for all . I will first count the number of permutations with for some . For any one of those permutations, we have
I believe the count for the number of permutations with is:
Is this correct? I want to make sure I am counting correctly before I continue.