Proving ##\frac{n!}{r(n-r)!}## Distinct r-Cycles in $S_n$

[Lee33]

Homework Statement

In $S_n$, prove that there are $\frac{n!}{r(n-r)!}$ distinct r-cycles.

2. The attempt at a solution

I know there are n choose r ways to permute r out of n cycles thus $\frac{n!}{r!(n-r)!}$ but I don't know how they got $\frac{n!}{r(n-r)!}$?

 

[pasmith]

Homework Statement

In $S_n$, prove that there are $\frac{n!}{r(n-r)!}$ distinct r-cycles.

2. The attempt at a solution

I know there are n choose r ways to permute r out of n cycles thus $\frac{n!}{r!(n-r)!}$

Two r-cycles are the same if their cycle notations are cyclic permutations of each other. Having chosen our objects, we can avoid such multiple counting by fixing one object to appear the beginning of all the r-cycles. Each permutation of the remaining $r-1$ objects will then generate a distinct r-cycle.