# 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 ofexactly 3 girlsat 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 ${}^{4}C_{3}=4$ ways
and we can choose remaining 4 candidates from 9 boys in ${}^{9}C_{4}$ ways.
thus the number of ways in which committee has exactly 3 girls = ${}^{9}C_{4}*4=\frac{9*8*7*6}{4*3*2}*4=504$

(ii) committee has at most 3 girls.
total number of students = 9+4=13
total number of ways of selecting 7 members = ${}^{13}C_{7}=\frac{13*12*11*10*9*8}{6*5*4*3*2}=13*12*11=1716$
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
$={}^{4}C_{4}*{}^{9}C_{3}\phantom{\rule{0ex}{0ex}}=1*\frac{9*8*7}{3*2}=84$
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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