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

$2y=91x$

Explanation for the correct answer:

Given otal number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denoting total no. of one one funtion

Number of elements in $n\left(A\right)=3$

Number of elements in $n\left(B\right)=5$

Number of elements in $n(A\times B)=15$

We know that number of one-one functions$={}^q{C}_p\times p!$

The number of one-one functions is

$\begin{array}{rcl}x& =& {}^5{C}_3\times 3!\\ & =& \frac{5!}{3!2!}\times 3!\\ & =& 60\end{array}$

The number of one-one functions is

$\begin{array}{rcl}y& =& {}^{15}{C}_3\times 3!\\ & =& \frac{15!}{12!\times 3!}\times 3!\\ & =& 2730\end{array}$

Therefore,

$$\begin{gathered} \frac{x}{y}=\frac{60}{2730} \\ \Rightarrow 273x=6y \\ \Rightarrow 2y=91x \end{gathered}$$

Hence, the correct answer is option (B).