UPSC NDA Math Model Paper 3

The possibilities of selecting 7 people, such that at most 3 are girls, are as follows:
(7 boys AND 0 girls) OR (6 boys AND 1 girl) OR (5 boys AND 2 girls) OR (4 boys AND 3 girls)
Using the basic principle of counting, the total number of ways will be:
(‌9C7×‌4C0)+(‌9C6×‌4C1)+(‌9C5×‌4C2) +(‌9C4×‌4C3)
=(

9!

7!(9−7)!

×

4!

0!(4−0)!

) +(

9!

6!(9−6)!

×

4!

1!(4−1)!

) +(

9!

5!(9−5)!

×

4!

2!(4−2)!

) +(

9!

4!(9−4)!

×

4!

3!(4−3)!

)
=(36×1)+(84×4)+(126×6) +(126×4)
= 1632.