Question

Four couples (husband and wife) decided. Find the number of different committees that can be formed in which no couple finds a place.

• A. $12$
• B. $14$
• C. $16$
• D. $24$

Answer 1

The correct option is C

$16$

Explanation for the correct option:

Find the number of different committees :

There are four ways to form a committee of four-member

All are gentlemen, $3$husband, $1$wife, $2$husband, $2$wife, and all are wives.

The number of committees of $4$ gentlemen $\Rightarrow {}^{4}C_4=1$

The number of committees of $3$gentlemen, $1$wife $\Rightarrow {}^{4}C_3\times {}^{1}C_1=4$

The number of committees of $2$gentlemen, $2$wife$\Rightarrow {}^{4}C_2\times {}^{2}C_2=6$

The number of committees of $4$wives $\Rightarrow {}^{4}C_4=1$

The required number of committees $\Rightarrow 1+4+6+4+1=16$
The number of different committees that can be formed in which no couple finds a place is $16.$

Hence, the correct option is (C).