Question

A committee of 3 men and 4 women , is to be formed from 5 men and 7 women . How many different committees can be formed if :

Answer 1

Answer:

Part A

A particular man on the seat, then 2 men are out of 4:

• 4C2 = 4!/(4-2)!2! = 4*3*2/2*2 = 6

4 women out of 7:

• 7C4 = 7!/(7-4)!4! = 7*6*5*4!/3!4! = 35

Total combinations:

• 6*35 = 210

Part B

Option 1. 2 men excluded

3 men out of 3 and 4 women out of 7

Total ways:

• 7C4 = 35 (as above)

Option 2. 2 women are excluded

3 men out of 5 and 4 women out of 5

• 5C3*5C4 =
• 5!/3!(5-3)! * 5!/4!(5-4)! =
• 5*4/2 * 5/1 = 10

Option 3. 1 man and 1 woman excluded

3 out of 4 men and 5 out of 6 women:

• 4C3*6C5 =
• 4!/3!(4-3)! * 6!/5!(6-5)! =
• 4*6 = 24

Part C

No restrictions, so the number of ways:

• 3C5 * 7C4 =
• 5!/3!(5-3)! * 35 =
• 4*5/2 * 35 =
• 10*35 = 350