Question

There are 8 men and 10 women members of a club. How many committees of 7 can be formed having at least 4 women?

Answer 1

Number of men and women in the club: 8 men and 10 women

1) to find the number of ways the committe could be done with utmost 5 men.

This could done with commottee with (5men and 2women)or 4men and 3 women) or (3men and 4women)

So the combinations is: (8C5)(10C2)+(8C4)(10C3)+(8C3)(10C4)+(8C2)(10C5)+(8C1)(10C6)+(8C0)(10C7)

=18C7 - (8C7)(10C0) - (8C6)((10C1), as sum of mCi*nCr-i = (m+n)Cr , for i = 0 to r.

=31824 - 280 -8

= 30536

2)At least 3 women committe with 7 members could be:

(4men 3women) or (3men and 4women) or (2men and 5women) Or(1men and 6woen) or (0men and7women)

= 8C4*10C3 +8C3*10C4+8C2*10C5+8C1*10C6+8C0*10C7 ways

=18C7 -((8C5*10C2+8C6*10C1+8C7*10C0)

=31824-(1260+560+8)

=29996