Question

The number of surjective functions from $A$ to $B$ where $A=\{1,2,3,4\}$ and $B=\{a,b\}$ is

• A. $14$
• B. $12$
• C. $2$
• D. $15$

Answer 1

The correct option is A $14$

If $A$ and $B$ are two sets having $m$ and $n$ elements such that $1\le n\le m$

Then, no. of surjection $=\sum _{r=1}^n(-1{)}^{n-r}{}^n{C}_r{r}^m$
Number of surjection from $A$ to $B$
$=\sum _{r=1}^2(-1{)}^{2-r}{}^2{C}_r(r{)}^4$
$=(-1{)}^{2-1}{}^2{C}_1(1{)}^4+(-1{)}^{2-2}{}^2{C}_2(2{)}^4=-2+16$
$=14$

Hence, option A is correct.