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.