A group of 7 members having a majority of boys is to be formed out of 6 boys and 4 girls. the number of ways he group can be formed?

A group of 7 members having a majority of boys formed out of 6 boys and 4 girls :
(4 boys and 3 girls) or (5 boys and 2 girls) or (6 boys and 1 girl)

The number of ways to form the group
= ₆C₄ × ₄C₃ + ₆C₅ × ₄C₂ + ₆C₆ × ₄C₁
= 15 × 4 + 6 × 6 + 1 × 4
= 100

The number of boys is 4, 5 or 6.
The number of ways = the number of ways of choosing 4 boys and 3 girls + the number of ways of choosing 5 boys and 2 girls + the number of ways of choosing 6 boys and 1 girl = (6 C 4)(4 C 3) + (6 C 5)(4 C 2) + (6 C 6)(4 C 1) = (15 * 4) + (6 * 6) + (1 * 4) = 60 + 36 + 4 = 100.

7 members having a majority (4 or more) of boys...

4 boys, 3 girls...(6 C 4) * (4 C 3) = (15) * (4) = 60

OR

5 boys, 2 girls...(6 C 5) * (4 C 2) = (6) * (6) = 36

OR

6 boys, 1 girl...(6 C 6) * (4 C 1) = (1) * (4) = 4

100 WAYS !

It's marginally quicker to count a majority of girls in C(6,3)C(4,4) ways and subtract from total ways of choosing 7 children from 10 C(10,7):

C(10,7) - C(6,3)C(4,4)
= 10*9*8/(3*2) - 6*5*4/(3*2)
= 120 - 20
= 100