1. 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 $n$.
Solution:
4
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 $7!=n!\cdot 10!/(n!\cdot(10-$ $n)!)$. This is equivalent to $(10-n)!=10!/ 7!=8 \cdot 9 \cdot 10=720=6!$. We therefore have $n=4$.