UPSC NDA Math Model Paper 1

Given, A committee of 7 has to be formed from 9 boys and 4 girls.
A committee of at most 3 girls consists of
(1) No girl, 7 boys
(2) 1 girl, 6 boys
(3) 2 girls, 5 boys
(4) 3 girls, 4 boys
Total number of ways that consists of at most 3 girls =(‌4C0)(‌9C7)+(‌4C1)(‌9C6)+(‌4C2)(‌9C5) +(‌4C3)(‌9C4)
=(

4!

0!(4)!

)(

9!

2!(7)!

)+(

4!

3!(4)!

)(

9!

3!(6)!

) +(

4!

2!(2)!

)(

9!

4!5!

)+(

4!

1!(3)!

)(

9!

5!4!

).
=36+336+756+504
=1632
Hence, A committee of 7 has to be formed from 9 boys and 4 girls. total number of ways can this be done when the committee consists of at most 3 girls is 1632.