Permutation cycles My tasks are  the following :
Task 1 :
Prove that $  \begin{pmatrix} 1 & 2 & \cdots & r-1 & r \end{pmatrix}
=   \begin{pmatrix}  2 & 3&  \cdots & r & 1 \end{pmatrix} =   \begin{pmatrix} 3 & 4 & \cdots & 1 & 2  \end{pmatrix}=  \cdots  =   \begin{pmatrix} r & 1 &\cdots & r-1 \end{pmatrix}$
and conclude that there are exactly r such notations for a r-cycle 
My Attempt:
My understand of why this would be true as follows:
You are given a permutation $f \in S_X $ such that:
$f(1) =2 , f(2) =3 ,\ldots , f(r-1) = r , f(r) =1$
So when constructing a  r-cycle, we r choices for the first element in the cycle, while the remaining  $r -1 $  elements are dictated by $f$ and hence:
$r \times 1 \times 1 \times \cdots \times 1 = r $.
Task 2:
if $1 \leq r \leq n $ then there are $ \frac 1r [ n(n-1)\ldots (n-r +1)]$ r-cycles in $S_n$ 
My Attempt:
I again  have some understanding of why this is true. You have a set of $n$ elements. Of which  you want to find all possible r-cycles , this is the  number of permutations of the n elements  taken r element at a time divide by r i.e. the  number of way a cycle can be represented 
Can you offer any suggestions or tips on how to convert these rough ideas in fully fetched rigorous proofs?

You are almost done. Just put this into mathematical grammar (i.e. use the definition of a cycle by $(a_0, f(a_0), \ldots, f^r(a_0))$ and remark that $a_0$ is in a set of cardinality $r$ (namely the elements of the cycle of length $r$).
For Task 2, the argument is already formulated well. Just say that you are using the result from Task 1.
