In how many ways can a committee be formed of | vedclass

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Question

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

Vedclass · Accepted Answer

$246$

Vedclass · Answer

(B) The total number of ways to select $5$ members from $10$ people ($6$ men + $4$ women) is given by $^{10}C_5 = \frac{10 	imes 9 	imes 8 	imes 7 	imes 6}{5 	imes 4 	imes 3 	imes 2 	imes 1} = 252$.The number of ways to select a committee with no women (i.e.,all $5$ members are men) is given by $^6C_5 = 6$.The number of ways to form a committee with at least one woman is the total number of ways minus the number of ways with no women:Required ways $= ^{10}C_5 - ^6C_5 = 252 - 6 = 246$.Alternatively,by summing the cases:$1$ woman and $4$ men: $^4C_1 	imes ^6C_4 = 4 	imes 15 = 60$$2$ women and $3$ men: $^4C_2 	imes ^6C_3 = 6 	imes 20 = 120$$3$ women and $2$ men: $^4C_3 	imes ^6C_2 = 4 	imes 15 = 60$$4$ women and $1$ man: $^4C_4 	imes ^6C_1 = 1 	imes 6 = 6$Total ways $= 60 + 120 + 60 + 6 = 246$.

In how many ways can a committee be formed of $5$ members from $6$ men and $4$ women if the committee has at least one woman?

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