A class comprises 26 students. During a mathematics test, one student made 12 mistakes but everyone else in the class made fewer mistakes than he did. Prove that there were at least 3 students in the class who made the same number of mistakes.

During a mathematics test, one student made 12 mistakes but everyone else in the class made fewer mistakes than he did. So, all other students made 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 or 0 mistakes (12 different possibilites).

Suppose that at most 2 students in the class made the same number of mistakes. Hence, at most [tex]12\cdot 2=24[/tex] students satisfy this condition. If we add that student who made 12 mistakes, the maximum total number of students in the class could be

[tex]24+1=25[/tex] students.

But the total number of students is 26, so our hypothesis is incorrect, thus there are at least 3 students in the class who made the same number of mistakes.

A class comprises 26 students. During a mathematics test, one student made 12 mistakes but everyone else in the class made fewer mistakes than he did. Prove that there were at least 3 students in the class who made the same number of mistakes.​

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A class comprises 26 students. During a mathematics test, one student made 12 mistakes but everyone else in the class made fewer mistakes than he did. Prove that there were at least 3 students in the class who made the same number of mistakes.