The number of surjection from $$A = \left\{ {1,\,2,\,.....,\,n} \right\},\,n \geqslant 2{\text{ onto }}B = \left\{ {a,\,b} \right\}$$         is :

Solution :
We know that, if $$X$$ and $$Y$$ are any two finite sets having $$m$$ and $$n$$ elements respectively, where $$1 \leqslant n \leqslant m,$$   then the number of onto functions from $$X$$ to $$Y$$ is given by $$\sum\limits_{r = 1}^n {{{\left( { - 1} \right)}^{n - r}}\,{}^n{C_r}{r^m}.r = 1} $$
Thus, the number of surjective mapping is $$\sum\limits_{r = 1}^2 {{{\left( { - 1} \right)}^{2 - r}}\,{C_r}{r^n} = \left( {{2^n} - 2} \right)} $$