Question

A mapping is selected at random from the set of all the mappings of the set $A=\{1,2,....,n\}$ into itself. Find the probability that the mapping selected is an injection.

Answer 1

Injection is a one - one function.

$A\to A^{\prime }$

If I map $1^{st}$ element of A to anyone element of $A^{\prime }$

then $2^{nd}$ element of A will have (n-1) option for mapping.

but if it is not one-one function then every element

in A can be mapped to any element of $A^{\prime },$ they each

have n options

If f(x) is one-one $\to n(n-1)...=n!$

If f(x) is not one-one $\to n\times n...ntimes=(n{)}^n$

Probability = $\frac{n!}{(n{)}^n}$