Let A = {1, 2, 3, .......... n} and B = {a, b, c}, the number of functions from A to B that are onto is

The number of onto functions from the set $A$ ( hanving $n$ elements) into the set $B$( having $m$ elements)  is $\sum _{r=1}^n{\left(-1\right)}^{m-r}\left(m_{C_r}\right)r^n$

Given A = {1, 2, 3, .......... n} and B = {a, b, c}

Number of onto functions 

$\begin{array}{c}\sum _{r=1}^n{\left(-1\right)}^{m-r}{m}_{C_r}{r}^n={\left(-1\right)}^{3-1}\left(3_{C_1}\right){\left(1\right)}^n+{\left(-1\right)}^{3-2}\left(3_{C_2}\right){\left(2\right)}^n+{\left(-1\right)}^{3-3}\left(3_{C_3}\right){\left(3\right)}^n\\ =3-3\left(2^n\right)+3^n\\ =3^n-3\left(2^n-1\right)\end{array}$

Therefore, the number of onto functions is $\boxed{3^n-3\left(2^n-1\right)}$