A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done

Question

asked May 4, 2024 42.6k views

1 Answer

Kiran Babu · Answer 1 · 2024-05-08T11:55:04+0000

Work Shown

There are n = 9+4 = 13 people to pick from, and there are r = 7 seats on the committee.

We cannot repeat a selection. Order does not matter so we use the nCr combination formula.

n C r = (n!)/(r!(n-r)!)

13 C 7 = (13!)/(7!*(13-7)!)

13 C 7 = (13!)/(7!*6!)

13 C 7 = (13*12*11*10*9*8*7!)/(7!*6!)

13 C 7 = (13*12*11*10*9*8)/(6!)

13 C 7 = (13*12*11*10*9*8)/(6*5*4*3*2*1)

13 C 7 = 1235520/720

13 C 7 = 1716

There are 1716 ways to form this committee.

Side notes:

If the seats on the committee were named (eg: president, vp, etc), then the order would matter.
The value 1716 can be found in Pascal's Triangle. Look at the row that starts with 1, 13, ... to locate the value 1716. This is the 8th item in the list (recall that Pascal's Triangle starts the index at r = 0).
This committee could consist of all boys or a mix of both genders. It's not possible to have all girls because there are 4 girls and 7 seats.