Q:  Show that if seven integers are chosen from the first 10 positive integers, there must be at least two pairs with the sum of 11.

A:  “Show” is such a loose term.  Show by proof?  Show by logic?  Are these both the same essentially?  Here is my initial thought:

The first positive 10 integers are the numbers 1-10.  How many pairs sum to 11?

(1, 10), (2, 9), (3, 8), (4, 7), (5, 6)

There are 5 pairs that sum to 11.

If you are chosing 7 numbers and leaving out 3 numbers, what is the worst-case-scenerio?  You leave out one number from each pair.  So, that leaves out 3 of the 5 pairs, leaving 2 remaining pairs.