TITLE: How many ways to combine $7$ women and $6$ men in a committee of $6$ given each sex must be represented.
QUESTION [0 upvotes]: I thought something along the lines of: Number of ways to pick out $1$ woman from seven times the number of ways to pick out $1$ man of $6$ times the numbers of ways to arrange the rest $11$ and $4$.  $7\choose 1$ $6\choose1$ $11\choose4$ $= 13860$.
But this is not correct...

REPLY [0 votes]: Let M be the men,
and W be the women.
 number of ways to get a committee of 6, so that each gender is represented is
5W + M,
4W + 2M, 
3W +3M, 
2W + 4M,
W + 5M, 
which is = 7C1 * 6C5 + 6C2*5C4 +5C3*4C3+4C4 *3C2 + 2C1
= 7*6 + 15*5 + 10*4 + 1*3 + 2
 = 42 + 75 + 40 + 3 + 2
162 ways
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