Help me to prove this,
Show that if and with then .
For each $\displaystyle r\in\{1,\ldots,n\}$, let $\displaystyle X_r$ be the set of all permutations in $\displaystyle S_n$ which fix r. Then $\displaystyle X_r$ is a set of permutations on n-1 numbers, so $\displaystyle |X_r|=(n-1)!$ - hence $\displaystyle |S_n:X_r|=n$, and there is a one-to-one correspondence between $\displaystyle X_r$ and $\displaystyle S_{n-1}$