Question

Let $f\left(x\right)$ be a function defined from $A$ to $B$ where number of elements in $A$ and $B$ are $3$ and $7$ respectively. Then which among the following is/are correct.

• A. Number of many-one functions from $B$ to $A$ is $3^7$
• B. Total number of functions that can be defined from $B$ to $A$ is $3^7$
• C. Total number of functions that can be defined from $A$ to $B$ is $7^3$
• D. Number of one-one functions from $A$ to $B$ is $210$

Answer 1

Answer: (A), (B), (C), (D)

Number of one-one functions from $A$ to $B$ is $210$

We know that if $n\left(A\right)=m$ and $n\left(B\right)=n$ then,
$\left(i\right)$ Total number of functions that can be defined from $A$ to $B=n^m$
$\left(ii\right)$ Number of one one functions from $A$ to $B$ $={}^n{P}_m$
$\left(iii\right)$ Number of many one function's
$=$ Total number of functions $-$ Number of one-one functions

So, from $A$ to $B$,
$\left(i\right)$ Total number of functions $=7^3=343$
$\left(ii\right)$ Total number of one-one functions $={}^7{P}_3=210$
$\left(iii\right)$ Total number of many-one functions $=343-210=133$

From $B$ to $A$,
$\left(i\right)$ Total number of functions $=3^7$
$\left(ii\right)$ Total number of one-one functions $=0$
$\left(iii\right)$ Total number of many-one functions $=3^7$