## bahrom7893 4 years ago 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?

1. bahrom7893

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

2. anonymous

multiply them then multiply by 5!

3. bahrom7893

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

4. bahrom7893

lol that kinda came out funny..

5. 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

6. bahrom7893

Actually that one's not really clear either.

7. anonymous

oh ok then lets go slower

8. 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

9. bahrom7893

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

10. anonymous

similarly we can compute 12 choose 5 easily enough

11. 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

12. bahrom7893

ahh ok

13. 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

14. 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

15. bahrom7893

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

16. anonymous

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

17. anonymous

and that is of course 5!

18. anonymous

19. anonymous

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

20. bahrom7893

the 5!* that mess is slowly sinking in

21. 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

22. anonymous

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

23. bahrom7893

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

24. bahrom7893

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

25. 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

26. bahrom7893

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

27. anonymous

if i can, i will help