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Problem 799

AMC 12 late, AIME early

Combinatorics Difficulty 4.4 Find the answer Harvard-MIT Mathematics Tournament · United States

10 people are playing musical chairs with $n$ chairs in a circle. They can be seated in $7!$ ways (assuming only one person fits on each chair, of course), where different arrangements of the same people on chairs, even rotations, are considered different. Find $n$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Solution:

The number of ways 10 people can be seated on $n$ chairs is $n!$ multiplied by the number of ways one can choose $n$ people out of 10. Hence we must solve $7! = n! \cdot \dfrac{10!}{n! \cdot (10-n)!}$ . This is equivalent to $(10-n)! = \dfrac{10!}{7!} = 8 \cdot 9 \cdot 10 = 720 = 6!$ . We therefore have $n = 4$ .

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