Number of onto mappings from set {1,2,3,4,5} to the set {a,b,c}

This is based on the question at here

I want to know how many onto functions are there from set $A={1,2,3,4,5}$ to set $B={a,b,c}$.

This is how I did it: First i tried to find the functions which are not onto
case 1: All in A maps to a single element of B
There are 3 ways for this

case2:
When mappings are made for only two elements in range.