Question

In a club with 8 male and 10 female​ members, how many 4​-member committees can be chosen that have

​(a) at least 3 ​women?

​(b) no more than 2 ​men?

Answer 1

Number of Male members in a club = 8

Number of Female members in a club = 10

(a) Number of 4-member committees can be choosen that have

atleast 3 women;  

Option 1 : 3 women and 1 man ;

Number of ways of selecting 3 women from 10 women =

Number of ways of selecting 1 man from 8 men =

Therefore Number of ways of selecting 3 women from 10 women and selecting 1 man from 8 men =120x8 = 960

Option 2 : All 4-members are Women

Number of ways of selecting 4 women from 10 women =

Adding number of ways in option 1 and option 2 = 960 + 210 = 1170

Number of 4-member committees can be choosen that have atleast 3 women = 1170

(b) Number of 4-member committees can be choosen that have

No more than 2 men; Three options to have a 4-member committee with no more than 2 men: (0 men, 4 women), ( 1 man , 3 women) , (2 men, 2 women)

Option 1 : (0 man, 4 women)

All 4-members are Women

Number of ways of selecting 4 women from 10 women =

Option 2 :

3 women and 1 man ;

Number of ways of selecting 3 women from 10 women =

Number of ways of selecting 1 man from 8 men =

Therefore Number of ways of selecting 3 women from 10 women and selecting 1 man from 8 men =120x8 = 960

Option 3 :

2 women and 2 men ;

Number of ways of selecting 2 women from 10 women =

Number of ways of selecting 2 men from 8 men =

Therefore Number of ways of selecting 2women from 10 women and selecting 2 men from 8 men =45x28 = 1260

Adding number of ways in option 1 , option 2 and option 3= 960 + 210 +1260 = 2430

Number of 4-member committees can be choosen that have no more than 2 men = 2430