A four-member committee is to be formed from | vedclass

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(A) The total number of people is $9 + 5 = 14$. We need to form a committee of $4$ members. The total number of ways to select $4$ members from $14$ i

Vedclass · Accepted Answer

$\frac{125}{143}$

Vedclass · Answer

(A) The total number of people is $9 + 5 = 14$. We need to form a committee of $4$ members. The total number of ways to select $4$ members from $14$ is given by $^{14}C_4 = \frac{14 	imes 13 	imes 12 	imes 11}{4 	imes 3 	imes 2 	imes 1} = 1001$. We want the probability that the committee contains at least one woman. It is easier to calculate the complement: the probability that the committee contains no women (i.e.,all $4$ members are men). The number of ways to select $4$ men from $9$ men is $^{9}C_4 = \frac{9 	imes 8 	imes 7 	imes 6}{4 	imes 3 	imes 2 	imes 1} = 126$. The probability of selecting no women is $P(	ext{No women}) = \frac{126}{1001} = \frac{18}{143}$. The probability of selecting at least one woman is $1 - P(	ext{No women}) = 1 - \frac{18}{143} = \frac{125}{143}$.

$A$ four-member committee is to be formed from a group containing $9$ men and $5$ women. If a committee is formed randomly,then the probability that it contains at least one woman is

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