Number of conjugates in symmetric group

Question

I have the permutation $\sigma=(123)$ in the symmetrical group $S_5$ of degree $5$ and I want to find the number of conjugates to sigma in $S_5$. I know that the conjugates to sigma will be the permutations, which has the same cycle-type as sigma. So I want to count the number of permutations in $S_5$, that is a $3$-cycle. I just get a $3$-cycle by picking out $3$ elements of $\{1 2 3 4 5\}$. However, I've been told that the answer should be $20$. I don't understand the calculation I need to do in order to find that there are $20$ permutations in $S_5$ with the same cycle-type as sigma?

Answer 1

Write your cycle $(a,b,c)$.

There are $5$ choices for $a$, there are $4$ choices for $b$ and $3$ choices for $c$. Thus, there are $5.4.3 = 60$ ways to choose cycles $(a,b,c)$.

However, we counted every element $3$ times this way!

for example $$(1,2,3) = (2,3,1) = (3,1,2)$$ three times

So, the final answer is $60/3 = 20$.

Answer 2

There are $5\times4\times3$ ways to choose $3$ different elements from {$1,2,3,4,5$} -- $5$ choices for the first, $4$ choices remaining for the second, and $3$ choices remaining for the third -- but then you also have to consider that the $3$ cycles $(i_1i_2i_3)$, $(i_2i_3i_1)$, and $(i_3i_1i_2)$ are the same permutation, so divide by $3$ to get the answer you're looking for.