Question

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A\times B$. Then:

• A. $y=273x$
• B. $2y=91x$
• C. $y=91x$
• D. $2y=273x$

Answer 1

The correct option is B $2y=91x$

Given,
$n\left(A\right)=3,n\left(B\right)=5$
$\Rightarrow n(A\times B)=15$

Number of one-one functions from $A$ to $B$ is $x=5\times 4\times 3$
$\Rightarrow x=60$

Number of one-one function from $A$ to $A\times B$ is $y=15\times 14\times 13$
$\Rightarrow y=15\times 4\times \frac{14}{4}\times 13$
$\Rightarrow y=60\times \frac{7}{2}\times 13$
$\Rightarrow 2y=\left(13\right)\left(7x\right)\Rightarrow 2y=91x$