Question

Twelve people join hands for a circle dance. Suppose 6 are men and the other 6 are women. If they alternate by gender, in how many ways can they do the dance?

Answer 1

the men and women cannot be distinguished from one another, yes/

Answer 2

Is there some sort of combination or permutation I can use to solve this?

Answer 3

yes. circular permutation

Answer 4

\[P = (n-1)!\]
where n = number of entities

Answer 5

in this case, i suggest you count one pair as one entity

Answer 6

then count the number of pairs...then use circular permutation

Answer 7

then multiply by 2 (because the order of the pairs are interchangeable0

Answer 8

i suppose you did not get what i said.../

Answer 9

I'm working it out, just hold on

Answer 10

oh good. i thought my multiple comments confused you

Answer 11

So I'm getting that there are 6! possible pairs of males and females

Answer 12

there are 6 pairs.... but the circular permutation is (n-1)!

Answer 13

in this case, your n is 6 (because you have 6 pairs, thus, 6 entities

Answer 14

So I have 6 pairs, thus (6-1)! = 5!

Answer 15

right. then, since the pairs are interchangeable in order, you multiply by 2!

Answer 16

so I get 5!2!, but why am I multiplying by 2! ?

Answer 17

because the pairs are interchangeable

think of it like this

for example, this is one order

b g b g b g

^think of them as in a circle

another order would be

g b g b g b

see what i mean?

Answer 18

ok I see what you mean

Answer 19

good.

Answer 20

so that would be the end