Permutations And Combinations

Question

# A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has:(i) no girl (ii) at least one boy and one girl (iii) at least 3 girls.

Solution

Total number of boys and girls in the group = 4 girls + 7 boys = 11
Number of boys and girls in the team = 5
(i)      The team consists of no girl,
The team consists of 0 girl + 5 boys
∴        Number of selections =
Hence, the number of teams formed = 21
(ii) The team consists of at least 1 boy and 1 girl.
Options are:
The team consists of 1 girl + 4 boys
Number of selections =
= 4 x 7 x 5 = 140.
Or
The team consists of 2 girls + 3 boys
Number of selections =
=
Or
The team consists of 3 girls + 2 boys
Number of selections =
=
Or
The team consists of 4 girls + 1 boy.
Number of selections =
Hence, the total number of teams that can be formed = 140 + 210 + 84 + 7 = 441
(iii) The team consists of at least 3 girls
Options are:
The team consists of 3 girls + 2 boys.
Number of selections =
Or
The team consists of 4 girls + 1 boy.
Number of selections =
Hence, the total number of teams that can be formed = 84 + 7 = 91