Question

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

• A. $\frac { 1 } { n ^ { n } }$
• B. $\frac { 1 } { n ! }$
• C. $\frac { ( n - 1 ) ! } { n ^ { n - 1 } }$
• D. $\frac { n ! } { n ^ { n - 1 } }$

Answer 1

Answer: (C)

$\frac { ( n - 1 ) ! } { n ^ { n - 1 } }$

Answer 2

The total number of functions from $A$ to itself is $n ^ { n }$ and the total number of bijections from $A$ to itself is $n !$ {Since $A$ is a finite set, therefore every injective map from $A$ to itself is bijective also}.

$\therefore$The required probability $= \frac { n ! } { n ^ { n } } = \frac { ( n - 1 ) ! } { n ^ { n - 1 } }$