Question

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?

Answer 1

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

Answer 2

multiply them
then multiply by 5!

Answer 3

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

Answer 4

lol that kinda came out funny..

Answer 5

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

Answer 6

Actually that one's not really clear either.

Answer 7

oh ok then lets go slower

Answer 8

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

Answer 9

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

Answer 10

similarly we can compute 12 choose 5 easily enough

Answer 11

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

Answer 12

ahh ok

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

and that is of course 5!

Answer 18

so you multiply all this mess together to get your answer

Answer 19

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

Answer 20

the 5!* that mess is slowly sinking in

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

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

Answer 27

if i can, i will help