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
$\Rightarrow \,{}^4C_1 \times \,{}^4C_2 = 24$
(B) $B$ excluded
$\Rightarrow \,{}^4C_1 \times \,{}^5C_3 = 40$
(II) Miss $C$ is not taken
$\Rightarrow B$ does not come
$\Rightarrow \,{}^4C_2 \times \,{}^5C_3 = 60$
$\Rightarrow $ Total $=124$
Alternate method:
Case I:
Mr. $B$ is present
$\Rightarrow A$ is excluded and $C$ included
Hence, the number of ways is $^4C_2 \,{}^4C_1 = 24$.
Case II:
Mr. ‘$B$’ is absent
$\Rightarrow $ No constraint
Hence, the number of ways is $^5C_3\,{}^5C_2 = 100$.
$\therefore $ Total $=124$.