Of a group of twelve students 7 male 5 female How ways can I

Of a group of twelve students, 7 male 5 female. How ways can I pick 4 students:

a.     How many ways can I appoint a czar and assistant czar from this group of students?

b.     How many ways can pick a committee of 3 students consisting of 1 male and 2 female?

c.     How many ways can I choose five students given that at least 3 must be female?

Solution

There are totally 12 students, 7 male, 5 female

4 students are to be picked from them

a) Appointing a Czar can be in any one of 12 ways from 12 students

and assistant remaining 11 ways

= 12x11 = 132 ways

b) 1 male can be selected from 7 males in 7 ways

2 females can be selected from 5 females in 5C2 = 10 ways

Hence total no of ways = 7(10) =70 ways

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c) 5 students to be chosen atleast 3 female means

(3f, 2m) or (4f, 1m) or (5f, 0m)

There is no other way to select atleast 3 females

No of ways for 3f and 2m = 5C2 7C3 =350 ways

4f and 1m                      = 5C4 7C1 = 35 ways

5f                                  = 1 way

Total no of ways = 350+35+1 = 386