Question

By using the pigeonhole principle, let s = 3, 7, 11, 15t, 19, ___________ 95, 99, 103. How many elements must we select from s to ensure that there will be at least 2 whose sum is 110?

Answer 1

To ensure that there will be at least two elements whose sum is 110 from the set s, we need to select at least 111 elements from s.

Step-by-step explanation:

To apply the pigeonhole principle in this problem, we need to consider the sum of pairs of elements from the set s. The largest possible sum of two elements from s is 103 + 99 = 202. Since we want to ensure at least two elements sum to 110, we need to consider the sums from 1 to 110, which is a total of 110 pairs. There are 17 elements in s. Using the pigeonhole principle, we can say that at least one pair must sum to 110 if we select 110+1=111 elements from s.