Soccer Team

Book: A Walk through Combinatorics

Problem

A soccer team scored total of 40 goals. 9 players scored at least one goal. Prove that there are at least two players who scored the same number of goals.

Proof

Proof by contradiction

If each of the 9 players have a unique number of goals, then the set should be at the minimum $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

Sum of the numbers in the set is $45$ which is higher than $40$, the number of goals. So, our initial assumption that each player has a unique number of goals is false.

Hence, there must be at least two players among the 9 who have the same number of goals.