Question

Let $A$ and $B$ be two finite sets having $m$ and $n$ elements respectively such that $m\le n$. A mapping is selected at random from the set of all mappings from $A$ to $B$. The probability that the mapping selected is an injection, is

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

Answer 1

Answer: (A)

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

A has m elements and B has n elements

Total number of mapping from A to B =$m^n$

Injection mapping every element of A will be having distinct value in B so it is the total number of ways to choose m elements from n elements with permutation=${n_C}_m$*m!

Probability=${n_C}_m$*m!/$m^n$=$\frac{n!}{(n-m)!n^m}$