<s>[INST] How many committees of $7$ people can be formed in a class of $14$ boys and $10$ girls if the committee is to contain at least $4$ boys? In a class of $14$ boys and $10$ girls, a committee of $7$ is to be formed. How many committees are possible.
(a) if anybody can serve in the committee?
(b) if the committee is to have exactly $4$ boys?
(c) if the committee is to contain at least $4$ boys?
 [/INST] Case 1- 
C(24,7) 
= $\frac{24!}{7!\times17!}$
Case 2-
C(14,4) $\times$ C(10,3)
= $\frac{14!}{4!\times10!} \times \frac{14!}{4!\times10!}$
Case 3- 
At least 4 boys, so may be 5 or 6 or 7.
C(14,4) $\times$ C(10,3) + C(14,5) $\times$ C(10,2) + C(14,6) $\times$ C(10,1) + C(14,4)
= $\frac{14!}{4!\times10!} \times \frac{10!}{3!\times7!} + \frac{14!}{5!\times9!} \times \frac{10!}{2!\times8!} + \frac{14!}{6!\times8!} \times \frac{10!}{1!\times9!} + \frac{14!}{7!\times7!}$
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