Question

The total number of injective mappings from a set with $m$ elements to a set with $n$ elements,$m\le n,$ is

• A. $m^n$
• B. $n^m$
• C. $\frac{n!}{(n-m)!}$
• D. $n!$

Answer 1

Answer: (C)

$\frac{n!}{(n-m)!}$

Let $A=\{a_1,a_2,a_3.....a_m\}$
and $B=\{b_1,b_2,b_3.....b_n\}$ where $m\le n$
Given $f:A\to B$ be an injective mapping.
So, for $a_1\in A$, there are $n$ possible choices for $f\left(a_1\right)\in B$.
For $a_2\in A$, there are $(n-1)$ possible choices for $f\left(a_2\right)\in B$.
Similarly for $a_m\in A$, there are $(n-m-1)$ choices for $f\left(a_m\right)\in B$
So, there are $n(n-1)(n-2).....(n-m-1)=\frac{n!}{(n-m)!}$ injective mapping from $A$ to $B.$