Question

A committee of 12 is to be formed from 9 women and 8 men in which at least 5 women have to be included in a committee. Then the numbers of committees in which the women are in majority and men are in majority are respectively.

• A. 4784, 1008
• B. 2702, 3360
• C. 6062, 2702
• D. 2702, 1008

Answer 1

The correct option is D

2702, 1008

We can select at least 5 women from following ways

Case1: 5 women + 7 men

Case2: 6 women + 6 men

Case3: 7 women + 5 men

Case4: 8 women + 4 men

Case5: 9 women + 3 men


The number of ways in which at least 5 women can be included in a committee is

${}^9{C}_5\times {}^8{C}_7+{}^9{C}_6\times {}^8{C}_6+{}^9{C}_7\times {}^8{C}_5+{}^9{C}_8\times {}^8{C}_4+{}^9{C}_9\times {}^8{C}_3$

= 1008 + 2352 + 2016 + 630 + 56 = 6062 Ways

(i) Cases in which women are in majority

Case3: 7 women + 5 men

Case4: 8 women + 4 men

Case5: 9 women + 3 men

Sum of all these cases =

${}^9{C}_7\times {}^8{C}_5+{}^9{C}_8\times {}^8{C}_4+{}^9{C}_9\times {}^8{C}_3$