If $A = \{ 1,2,3, \ldots ,n\}$ and $B = \{ a,b\}$.

Then, the number of surjections from $A$ into $B$ is

It is given that,, $A = \{ 1,2,3, \ldots ,n\}$ and $B = \{ a,b\}$.

We know that, if $A$ and $B$ are two non-empty

finite sets containing $m$ and $n$ elements

respectively, then the number of surjection

from $A$ into $B$ is

$^n{C_m} \times m!$, if $n \ge m$

$0$, if $n < m$

Here, $m = 2$

$\therefore$

Number of surjection from $A$ into $B$ is given by

$^n{C_2} \times 2! = \frac{{n!}}{{2!(n - 2)!}} \times 2!$

$= \frac{{n(n - 1)(n - 2)!}}{{2 \times 1(n - 2)}} \times 2! = {n^2} - n$