TITLE: How many cycles of length $k$ in $S_n$?
QUESTION [6 upvotes]: In the symmetrical group $S_n$, how many cycles of length $k$ can we form? 
After some research I am tempted to say $\frac{n!}k$ but I am not sure.

REPLY [7 votes]: Hints:

A cycle of length $k$ permutes $k$ elements and leaves the remaining $n-k$ alone. How many ways are they of choosing which $k$ elements the cycle permutes?
Once we've chosen $k$ elements to permute, we need to work out how many ways there are to permute them. Suppose your $k$ elements are $a_1, \ldots, a_k$. $a_1$ can go to any of $a_2, \ldots,  a_k$ - say it goes to $a_i$. Then $a_i$ can go to any of $a_2, \ldots, a_{i-1}, a_{i+1}, \ldots a_k$, and so on. How many possible permutations are there? Try some small values of $k$ if you get stuck.