Question

Let $S=\{1,2,3,4,5,6\}$. Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $g\left(3\right)=2g\left(1\right)$ is

• A. $\frac{1}{30}$
• B. $\frac{1}{10}$
• C. $\frac{1}{15}$
• D. $\frac{1}{5}$

Answer 1

The correct option is B $\frac{1}{10}$

Given: $S=\{1,2,3,4,5,6\}$.
Total number of onto funnctions $=6!$
$\because g\left(3\right)=2g\left(1\right)$
$\therefore \left(g\right(1),g(3\left)\right)=(1,2)$ or $(2,4)$ or $(3,6)$
In each case number of onto functions $=4!$
Required probability $=\frac{3\cdot 4!}{6!}=\frac{1}{10}$