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:
- 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
