Question

Group A contains $12$ persons, Group B contains $15$ persons and Group C contains $n$ persons. Every person in one group shakes hands with every person in the other group exactly once. The total number of handshakes is $666$. Then find the value of $n$.

• A. $16$
• B. $25$
• C. $18$
• D. $20$

Answer 1

Answer: (C)

$18$

Given:

Group A has $12$ persons

Group B has $15$ persons

Group C has $n$ persons

Total number of handshakes $=666$ ..... $\left(1\right)$

Each person of group $A$ will shake hands with each person of group $B$ and $C$ therefore there will be $12\times (15+n)$ hand shakes.

Also Group $B$ and $C$ also shake hands and hence numbers of handshakes of these people $=15\times n$

Hence total number of handshakes $=12\times (15+n)+15\times n$ ..... $\left(2\right)$

Equating $\left(1\right)$ and $\left(2\right)$

$12\times (15+n)+15\times n=666$

$180+27n=666$

$27n=486$

$n=18$

Hence, the correct answer is $18$.