Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 1E / i / Solution

A transposition exchanges exactly two letters and fixes the rest. A permutation cycle $(a_1{a}_2\dots ,a_k)$ sends each displayed letter to the next, sends $a_k$ to $a_1$, and fixes all other letters.

After imposing $\sigma (1)=2$, the remaining $n-1$ domain elements may be sent bijectively to the remaining $n-1$ values in arbitrary order. Hence the number is