bahrom7893
  • 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?
Mathematics
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SOLVED
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chestercat
  • chestercat
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bahrom7893
  • bahrom7893
Ok, so far I have 10C5 and 12C5, have no clue what to do next.
anonymous
  • anonymous
multiply them then multiply by 5!
bahrom7893
  • bahrom7893
why though? 5! is the permutation of 5 pairs... but why multiply men and women?

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More answers

bahrom7893
  • bahrom7893
lol that kinda came out funny..
anonymous
  • 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
  • bahrom7893
Actually that one's not really clear either.
anonymous
  • anonymous
oh ok then lets go slower
anonymous
  • 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
  • bahrom7893
yes, i know why we're doing 10C5 and 12C5, but why are we multiplying them together?
anonymous
  • anonymous
similarly we can compute 12 choose 5 easily enough
anonymous
  • 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
  • bahrom7893
ahh ok
anonymous
  • 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
  • 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
  • bahrom7893
ohhh i see.. geezz finally.. that bugged me for like 5 hours today..
anonymous
  • anonymous
that is like asking how many ways can you put five men in five chairs (not to be too crude about it)
anonymous
  • anonymous
and that is of course 5!
anonymous
  • anonymous
so you multiply all this mess together to get your answer
anonymous
  • anonymous
clear ? that counting principle, simple as it is, is powerful stuff
bahrom7893
  • bahrom7893
the 5!* that mess is slowly sinking in
anonymous
  • 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
  • anonymous
or vice versa if you don't want to be sexist about it
bahrom7893
  • bahrom7893
no I understand 5!, I mean 5! * everything else
bahrom7893
  • bahrom7893
this counting principle's always bugging me out... dang it.
anonymous
  • 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
  • bahrom7893
ok that makes it a little clearer. hold on, i have another question, for some reason they're adding this time.
anonymous
  • anonymous
if i can, i will help

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