# Conjugates in symmetric groups

#### kimberu

1. The problem statement, all variables and given/known data
The question is, "How many conjugates does (1,2,3,4) have in S7?

Another similar one -- how many does (1,2,3) have in S5?

3. The attempt at a solution
I know that the conjugates are all the elements with the same cycle structure, so for (123) I found there are 20 conjugates by hand listing them (132)...(354) etc. But I was wondering if there's some equation to find this amount- I haven't taken probability but I think there's got to be a statistical way to figure it out!! Thanks a lot.

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#### The Chaz

the binomial coefficient might be what you're looking for.
"nCr"
"choose function"

#### kimberu

using the formula on (1,2,3) in S5, I got that it has 10 conjugates, which is wrong -- which N and R should I use for this example if not 5 and 3 (or am I calculating wrong)?

#### The Chaz

No, you're doing it right (i.e. "this isn't what you're looking for). I forgot to divide by 2! (!)... :/

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