(B) Total number of ways to select $5$ members from $10$ people ($6$ men + $4$ women) is ${}^{10}C_5 = \frac{10 \times 9 \times 8 \times 7 \times 6}{5

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(B) Total number of ways to select $5$ members from $10$ people ($6$ men + $4$ women) is ${}^{10}C_5 = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252$.The number of ways to select a committee with no women (i.e.,all $5$ members are men) is ${}^6C_5 = 6$.The number of ways to select at least one woman is (Total ways) - (Ways with no women) = $252 - 6 = 246$.The required probability is $P = \frac{246}{252} = \frac{41}{42}$.

Class 11 Mathematics Probability Advanced Use of permutations and combinations in probability

Medium

$A$ committee of $5$ members is to be formed from $6$ men and $4$ women. What is the probability that the committee contains at least one woman?

A
$\frac{1}{42}$
B
$\frac{41}{42}$
C
$\frac{2}{63}$
D
$\frac{1}{7}$

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Advanced Use of permutations and combinations in probability

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$A$ committee of $5$ members is to be formed from $6$ men and $4$ women. What is the probability that the committee contains at least one woman?

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