A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of

exactly 3 girls

at most 3 girls

there are 9 boys and 4 girls. committee has 7 students.
(i) committee has exactly 3 girls.
therefore we can choose 3 girls out of 4 girls in C34=4 ways
and we can choose remaining 4 candidates from 9 boys in C49 ways.
thus the number of ways in which committee has exactly 3 girls = C49*4=9*8*7*64*3*2*4=504

(ii) committee has at most 3 girls.
total number of students = 9+4=13
total number of ways of selecting 7 members = C713=13*12*11*10*9*86*5*4*3*2=13*12*11=1716
at most 3 girls , therefore committee has either 0,1,2,or 3 girls.
the number of ways that committee has girls more than 3 is 4 girls and 3 boys
=C44*C39=1*9*8*73*2=84
thus the number of ways that committee has at most 3 girls 
= total number of ways - the number of ways that committee has girls more than 3.
= 1716 - 84
=1632

hope this helps you.

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