How to solve this Pigeon-hole Problem.

Question

show that there must be at least 90 ways to choose six number from 1 to 15 so that all choices have same sum.

Thanks.

Answer 1

You can choose 6 numbers from 15 in ${{15}\choose{6}}=5005$ ways and the possible sums are $75-21+1=55$ (75 is the max sum and 21 the least sum and we can make every sum between these numbers).

But $5005/55=91$ thus by pigeonhole we get the desired result.