A club consisting of 5 men and 8 women will choose a committee of 4. In how many ways can the committee be chosen if it must contain at least 1 man?

Question

A club consisting of 5 men and 8 women will choose a committee of 4. In how many ways can the committee be chosen if it must contain at least 1 man?

Answer 1

Pick the man first.  There are five possibilities for who can be picked.

 

After the man is picked, any of the remaining club members are valid selections for the remaining slots - this implies 12 possibilities (8 women plus the 4 remaining men) for the second slot, 11 for the third, and 10 for the fourth, for a total of 5*12*11*10=6600.

 

However, since this is just a committee of four and the slots are not actually distinguishable, order doesn't matter.  Therefore, we need to reduce this number of possibilities by the number of possible arrangements, which is just 4!.  (That is, four factorial, or 4*3*2*1, or 24)

 

The final number of ways is then 6600/24, or 275.