Question

1. A committee of 7 is to be formed out of 6 gents and 4 Ladies. In how many ways this can be done, when

               a). at least two ladies are included?                            b). at most two ladies are included?

Answer 1

It is given that a committee of 7 is to be formed out of 6 gents and 4 ladies.

a) Here, we have to find the number of ways the committee can be done when at least two ladies are included.

Thus, if a committee of 7 is to be formed and at least two ladies are included then we will have three cases.

Case 1 : When the committee consists of 2 ladies and 5 gents. Thus, we have to select 2 ladies from 4 ladies and 5 gents from 6 gents.

Thus, the number of ways =

We know,

= n!/{r!(n-r)!} where n!=n(n-1)(n-2)....1

Thus, = 6*6 = 36

Case 2 : When the committee consists of 3 ladies and 4 gents. Thus, we have to select 3 ladies from 4 ladies and 4 gents from 6 gents.

Thus, the number of ways =

= 4*15 = 60

Case 3 : When the committee consists of 4 ladies and 3 gents. Thus, we have to select 4 ladies from 4 ladies and 3 gents from 6 gents.

Thus, the number of ways = = 1*20 = 20

Thus, from the three cases, the total number of ways = 36 + 60 + 20 = 116

Thus, the total number of ways the committee can be done when at least two ladies are included = 116 .

b) Here, we have to find the number of ways the committee can be done when at most 2 ladies are included.

Thus, if a committee of 7 is to be formed and at most 2 ladies are included then we will have two cases.

Case 1 : When the committee consists of 1 lady and 6 gents. Thus, we have to select 1 lady from 4 ladies and 6 gents from 6 gents.

Thus, the number of ways = = 4*1 = 4

Case 2 : When the committee consists of 2 ladies and 5 gents. Thus, we have to select 2 ladies from 4 ladies and 5 gents from 6 gents.

Thus, the number of ways = = 6*6 = 36

Thus, from the two cases, the total number of ways = 4 + 36 = 40

Thus, the total number of ways the committee can be formed when at most 2 ladies are included = 40 .