- bahrom7893

A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible?

- jamiebookeater

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- bahrom7893

Ok, so far I have 10C5 and 12C5, have no clue what to do next.

- anonymous

multiply them
then multiply by 5!

- bahrom7893

why though? 5! is the permutation of 5 pairs... but why multiply men and women?

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## More answers

- bahrom7893

lol that kinda came out funny..

- anonymous

the first multiplication is clear yes? you have
\[\binom{10}{5}\] possibilities for the 5 women and
\[\binom{12}{5}\] for the five men. now we think about the pairings

- bahrom7893

Actually that one's not really clear either.

- anonymous

oh ok then lets go slower

- anonymous

first question is, how many ways can we pick five women from a set of 10 and that is just asking what is 10 choose 5, which we can compute

- bahrom7893

yes, i know why we're doing 10C5 and 12C5, but why are we multiplying them together?

- anonymous

similarly we can compute 12 choose 5 easily enough

- anonymous

by the "counting principle" if there are m ways to do one thing and n ways to do another, then there are mn ways of doing them together

- bahrom7893

ahh ok

- anonymous

think of it this way. for each group of 5 women (there are 252 possible groups) we can pair them up with each of the group of 5 men and there are 792 of them

- anonymous

now once we have selected one of our cominations of 5 men and 5 women, we want to see how many ways we can match them up

- bahrom7893

ohhh i see.. geezz finally.. that bugged me for like 5 hours today..

- anonymous

that is like asking how many ways can you put five men in five chairs (not to be too crude about it)

- anonymous

and that is of course 5!

- anonymous

so you multiply all this mess together to get your answer

- anonymous

clear ? that counting principle, simple as it is, is powerful stuff

- bahrom7893

the 5!* that mess is slowly sinking in

- anonymous

again it is the counting principle. you put the women in a row. how many choices of men for the first woman? 5
then he is matched, leaves 4 choices for the second women
etc
give
\[5\times 4\times 3\times 2\times 1=5!\] possible matches

- anonymous

or vice versa if you don't want to be sexist about it

- bahrom7893

no I understand 5!, I mean 5! * everything else

- bahrom7893

this counting principle's always bugging me out... dang it.

- anonymous

but clearer now i hope. counting principle again.
\[792\times 252\] possible groups of 5 men and 5 women, then once that is chosen another 5! ways to match them up

- bahrom7893

ok that makes it a little clearer. hold on, i have another question, for some reason they're adding this time.

- anonymous

if i can, i will help

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