Paper 2, Section I, E

What does it mean for an element of the symmetric group $S_{n}$ to be a transposition or a cycle?

Let $n \geqslant 4$. How many permutations $\sigma$ of $\{1,2, \ldots, n\}$ are there such that

(i) $\sigma(1)=2 ?$

(ii) $\sigma(k)$ is even for each even number $k$ ?

(iii) $\sigma$ is a 4-cycle?

(iv) $\sigma$ can be written as the product of two transpositions?

You should indicate in each case how you have derived your formula.

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