The Probabilistic Pigeon Hole Principle 2

Question

(a) A group of 15 boys plucked a total of 100 apples. Prove that two of those boys plucked the same number of apples.

Answer 1

Suppose not. Then, since a boy plucks at least $0$ apples, and the number of apples plucked by each boy is distinct. Then, the total number of apples plucked is at least $0 + 1 + ... + 14 = \frac{14(15)}{2} = 105 > 100$, a contradiction. Thus, there exists two boys who plucked the same number of apples.