Let A={1,2,...,5} and B={1,2,3} then how many onto functions to we have from A to B ?
Any one of the 5 elements of A can be mapped to any one element in B in 3 ways, Then one of the 4 elements of A to any 2 elements of B, and any 3 from A to one in B, therefore 3*2*1=6ways, now for the remaining 2 elements in A, One of them can be mapped to any three in B in 3 ways and the remaining one of A again to any 3 in B, Hence 3*3, Thus the total onto functions being 3!*3^2. And if this is correct then if |A| =m and |B|=n m=n then number of onto functions = n!*(n^(m-n)).
Is this right?

Question

Let A={1,2,...,5} and B={1,2,3} then how many onto functions to we have from A to B ?
Any one of the 5 elements of A can be mapped to any one element in B in 3 ways, Then one of the 4 elements of A to any 2 elements of B, and any 3 from A to one in B, therefore 3*2*1=6ways, now for the remaining 2 elements in A, One of them can be mapped to any three in B in 3 ways and the remaining one of A again to any 3 in B, Hence 3*3, Thus the total onto functions being 3!*3^2. And if this is correct then if |A| =m and |B|=n m>=n then number of onto functions = n!*(n^(m-n)). 
Is this right?