Permutations And Combinations

Question

# 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 :(i) exactly 3 girls? (ii) at least three girls? (iii) at most 3 girls?

Solution

Number of boys = 9
Number of girls = 4
The committee is to consist of 7
(i) The committee consists of exactly 3 girls
Number of boys = 7 - 3 = 4
∴              The number of selections =

=
(ii) The committee consists of at least 3 girls.
The committee consists of 3 girls + 4 boys or 4 girls + 3 boys
(Since, the number of girl is 4, so there is no further option)
When committee consists of 3 girls + 4 boys
Number of selections =
When committee consists of 4 girls + 3 boys
Number of selections =
Hence, the number of selections for the committee = 504 + 84 = 588.
(iii) The committee consists of at most 3 girls.
This gives us the following options:
No girl + 7 boys

Number of selections =
Or
1 girl + 6 boys
Number of selections =
Or
2 girls + 5 boys
Number of selections =

Or
3 girls + 4 boys
Number of selections =
Hence, the total number of selections = 36 + 336 + 756 + 504 = 1632.