How many ways to combine $7$ women and $6$ men in a committee of $6$ given each sex must be represented.

Question

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...

Answer 1

If only man are in commite we can do ${7\choose 6}$ commites and only one commite with only womans. So commites where only one gender is represented is $${7\choose 6}+1 =7+1=8$$

So we have $$ {13\choose 6}-8=...$$ commites.

Answer 2

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