Question

One mapping (function) is selected at random from all the mappings of set $A=\left\{1,2,3,..,n\right\}$ into itself. The probability that the mapping selected is one - one, is

• A. $\frac{n!}{n^n-1}$
• B. $\frac{n!}{n^n}$
• C. $\frac{n!}{2n^n}$
• D. None of these

Answer 1

The correct option is B

$\frac{n!}{n^n}$

Explanation for the correct option.

Finding the probability that the mapping selected is one - one, is

Given: One mapping (function) is selected at random from all the mappings of set $A=\left\{1,2,3,..,n\right\}$ into itself.

The total number of mapping from set $A=n^n$

The total number of one-one mapping is $n!$

Thus, the required probability will be the ratio of total number of one-one mapping to the total number of mapping from set $A$

$\begin{array}{rcl}P& =& \frac{n!}{n^n}\end{array}$

Hence the correct answer is option (B).