Question

The number of ways of choosing a committee of two women and three men from five women and six men, if Mr. A refuses to serve on the committee if Mr. $B$ is a member and Mr. $B$ can only serve, if Miss $C$ is the member of the committee is

• A. $60$
• B. $84$
• C. $124$
• D. none of these

Answer 1

Answer: (C)

(I) Miss $C$ is taken
(A) $B$ included
$\Rightarrow A$ excluded
$\Rightarrow {}^4{C}_1\times {}^4{C}_2=24$
(B) $B$ excluded
$\Rightarrow {}^4{C}_1\times {}^5{C}_3=40$
(II) Miss $C$ is not taken
$\Rightarrow B$ does not come
$\Rightarrow {}^4{C}_2\times {}^5{C}_3=60$
$\Rightarrow$ Total $=124$
Alternate method:
Case I:
Mr. $B$ is present
$\Rightarrow A$ is excluded and $C$ included
Hence, the number of ways is ${}^4{C}_2{}^4{C}_1=24$.
Case II:
Mr. ‘$B$’ is absent
$\Rightarrow$ No constraint
Hence, the number of ways is ${}^5{C}_3{}^5{C}_2=100$.
$\therefore$ Total $=124$.