Permutations and Combinations 2 Question 19

19. A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to be included in a committee? In how many of these committees

(i) the women are in majority?

(ii) the men are in majority?

(1994, 4M)

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Answer:

Correct Answer: 19. 6062 , (i) 2702 (ii) 1008

Solution:

  1. Given that, there are 9 women and 8 men, a committee of 12 is to be formed including at least 5 women.

This can be done in

$=(5$ women and $7 men)+(6$ women and $6 men)$

$+(7$ women and 5 men $)+(8$ women and 4 men $)$ + (9 women and 3 men) ways

Total number of ways of forming committee

$$ \begin{aligned} = & \left({ }^{9} C _5 \cdot{ }^{8} C _7\right)+\left({ }^{9} C _6 \cdot{ }^{8} C _6\right)+\left({ }^{9} C _7 \cdot{ }^{8} C _5\right) \\ & +\left({ }^{9} C _8 \cdot{ }^{8} C _4\right)+\left({ }^{9} C _9 \cdot{ }^{8} C _3\right) \\ = & 1008+2352+2016+630+56=6062 \end{aligned} $$

(i) The women are in majority $=2016+630+56$

$$ =2702 $$

(ii) The man are in majority $=1008$ ways



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