Question

This is a Combinatorics question.

Find a generating function for ar, the number of ways:

(A)

To distribute r identical objects into seven distinct boxes with an odd number of objects not exceeding nine in the first three boxes and between four and ten in the other boxes.

Answer 1

Let's make a bijection to strings because they're easier to count. Represent the situation as a string containing 8 "o" representing the balls and 2 "|" representing dividing the balls between the boxes. For example, 3 balls in box A, 3 in box B, and 2 in box C would be represented by "ooo|ooo|oo".

Since the problem states that none of the boxes can be empty, that means that you must have at least one 'o' (ball) in each box, which at the start would look like this:

" o | o | o "