The number of surjection from $\mathrm{A}=\{1,2,\dots \dots ,\mathrm{n}\},\mathrm{n}\ge$ onto B = {a, b} is

We know that, if X and Y are any two finite sets having m and n elements respectively, where $1\le n\le m$, then the number of onto functions from X to Y is given by $\sum _{f=1}^n {(-1{)}^{n-t}}^n{C}_t{r}^m\cdot r=1$

Thus, the number of surjective mappings is $\sum _{r=1}^2 (-1{)}^{2-r}{C}_r{r}^n=\left(2^n-2\right)$