how many different plate numbers can be made using the letters A,B and numbers 0,1,2,3 if each must contain 2 letters and 3 numbers.

Since you state no restriction, I will assume that the plate can start with a 0

first let's select our plate:
number of selections = C(2,2) C(4,3) = 4
then each of those can be arranged in 5! ways.

number of possible arrangements
= 4(5!) = 480