Question

How many Committees of five people Can be chosen from 20 men and 12 Women?

(i) If exactly three men must be on each Committee

(ii) If at least four women must be on each Committee​

Answer 1

Answer:

(i) 6

(ii) 3

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Answer 2

Answer:

1. 135498

2. 10692

Step-by-step explanation:

Given :-

• 20 men and 12 women

To find :-

no. of committees of 5 people each

Solution :-

Step 1)

• no. of men = 20
• no. of women = 12

Step 2) 3 men in one committee .

• since , we have 3 men on each committee , then no. of women will be 2
• no. of men = 3 , no. of women = 2
• By using combination formula ,

$2053* 66= 135498$ -- (1)

Step 3) At least 4 women or 5 women

• we will have sum of 2 combination .
• combination (1) , 4 women and 1 men ;

$^{20}C_1 * ^{12}C_4 \\= 20 * \frac{12*11*10*9}{4*3*2*1} \\= 20 * 495\\= 9900$--- (2)

• Combination (2) , 5 women and 0 men ;

$^{20}C_0 * ^{12}C_5\\= 1 * \frac{12*11*10*9*8}{5*4*3*2*1}\\= 1* 792 \\= 792$----- (3)

• summing up eqn. (2) and (3)

$= 9900+792= 10692 .$ --- (4)

Therefore , answer

1. 135498

2. 10692

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