Question

The circular permutations of $6$ different things taken all at a time is .....

Answer 1

The number of circular permutations of $n$ things taken $r$ at a time is $=\frac{{}^n{P}_r}{r}$

Here we are taking all the $n$ things at a time.

$\therefore$ No. of permutations $=\frac{{}^n{P}_r}{n}=\frac{n!}{n}$

$=(n-1)!$

In circular permutation, the first thing or the first place doesn't matter. So we place any of the $n$ things in any $1$ place and arrange the other $(6-1)$ things in $(6-1)!$ ways = $5!=120$ ways.