Question

In a beauty contest, half the number of experts voted for Miss A and two thirds voted for Miss B, $10$ voted for both and $6$ did not vote for either. Then how many experts were there in all?

• A. $20$
• B. $24$
• C. $22$
• D. $26$

Answer 1

Answer: (B)

$24$

Let the total number of experts be $N$.
$E$ is the set of experts who voted for miss $A$.
$F$ is the set of experts who voted for miss $B$.
Since $6$ did not vote for either, n$(E\cup F)=N-6$.
$n\left(E\right)=\frac{N}{2},n\left(F\right)=\frac{2}{3}N$ and $n(E\cap F)=10$
.
So, $N-6=\frac{N}{2}+\frac{2}{3}N-10$
Solving the above equation gives $\frac{N}{6}=4\Rightarrow N=24$