Pigeonhole Principle: Sum of $15$ positive integers is $100$, prove that two of them are the same

Question

Just confused as to how I'm suppose to set up the sets. Since there's an upper bound on each of the integers, could we use the Inclusion Exclusion Principle? I'm not too sure how to set that up because each integer depends on the one before it.

Anyway, any help would be appreciated. Thanks.

Answer 1

The smallest the sum of $15$ distinct, positive integers can be is

$$1+2+\ldots + 15 = \displaystyle{16\choose 2}=120>100.$$

So it must be that there are repeats in the list.