Fifteen 15 students take a quiz Their scores sum to 100 Prov

Fifteen 15 students take a quiz. Their scores sum to 100. Prove that there are two students that have the same score

Solution

Suppose the scores arrange them in order: n₁< n₂ < ... < n₁₅.

and let n₁ = 0, n₂ = 1 ..... n₁₅ = 14, then sum = 0 + 1 + 2 + ......+ 14 = 105.

This is a contradiction since the scores are supposed to sum to 100. Thus, the scores cannot be all different.

thus we can say that there are at least two students that have the same score.