Let $S_3$ be the symmetric group on $\lbrace 1,2,3\rbrace.$ Decide how many elements who commutate with $(23)$
Permutation naturally commutes itself, with it's inverse and with the identity permutation. So that's 3.
And then I'm insecure. What shall I do next?
First of all, the permutation $(23)$ is the inverse of itself so you actually found only $2$ permutations that commute with it so far, not $3$. Next, $S_3$ is a very small group, it has only $6$ elements. So you can just check every element in a direct way.