Answer
See step by step answer for solution.
Work Step by Step
$\underline{Pigeonhole}$ $ \underline{Principle}$- If k is a positive integer and k + 1 or more objects
are placed into k boxes, then there is at least one box containing two or more of the objects.
a) Consider the following tuples (1,10), (2,9), (3,8), (4,7), (5,6).
Look at the first entries of the tuples. These are five integers.
If we choose 6 integers, then by pigeon hole principle, there is at least one pair that add up to 11.
If we choose 5 integers, we may not get a pair that adds up to give 11 say {1,2,3,4,5}.
If we choose 7 integers, then by pigeon hole principle there are 2 pairs that add up to 11.
b) No this is not true. look at the set {1,2,3,4,5,6}. This contains only one pair that adds up to 11.