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

If A and B are two sets having m and n elements 

$\mathrm{Number} \mathrm{of} \mathrm{surjection} \mathrm{from} \mathrm{A} \mathrm{to} \mathrm{B}=\sum _{r=1}^n {(-1{)}^{n-r}}^n{C}_r{r}^m , 1\le n\le m$

Number of surjection from A to B 

$\begin{array}{l}=\sum _{\mathrm{r}=1}^{\mathrm{n}} {(-1{)}^{2-\mathrm{r}}}^2{\mathrm{C}}_{\mathrm{r}}(\mathrm{r}{)}^4\\ ={(-1{)}^{2-1}}^2{\mathrm{C}}_1(1{)}^4+{(-1{)}^{2-22}}^2{\mathrm{C}}_2(2{)}^4=-2+16=14\end{array}$