Question

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?

• A. $186$
• B. $246$
• C. $252$
• D. None of these

Answer 1

The correct option is B

$246$

Explanation for the correct option:

Finding the number of ways, a committee can be formed if the committee has at least one woman:

The possibilities for forming the committee are as follows:

Possibility $1$: $1$Woman $+$ $4$ Men $={}^4{C}_1\times {}^6{C}_4=60$

Possibility $2$: $2$ Women $+$ $3$ Men $={}^4{C}_2\times {}^6{C}_3=120$

Possibility $3$: $3$ Women $+$ $2$ Men $={}^4{C}_3\times {}^6{C}_2=60$

Possibility $4$: $4$ Women $+$$1$ Men $={}^4{C}_4\times {}^6{C}_1=6$

Thus, the total number of possible arrangements is $60+120+60+6=246$

Hence, option (B) is correct.