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

Answer: (A)

If $A$ and $B$ are two sets having $m$ and $n$ elements such that
$1\le n\le m=\sum _{r=1}^n(-1{)}^{n-r}{}^n{C}_r{r}^m$
Number of surjection from $A$ to $B$
$={\sum }_{r=1}^n(-1{)}^{2-r}{}^2{C}_r(r{)}^4$
$=(-1{)}^{2-12}{C}_1(1{)}^4+(-1{)}^{2-2}{}^2{C}_2(2{)}^4$
$=-2+16=14$