In a certain state, license plates each consist of 2 letters (from A to Z) followed by 3 digits (from 0 to 9). How many different license plates are there that have no repeated letters or digits?

Question

In a certain state, license plates each consist of 2 letters (from A to Z) followed by 3 digits (from 0 to 9). How many different license plates are there that have no repeated letters or digits? Write the number in the box below.

Answer 1

Since we are not allowed repetition, we simply decrease our multipliers by one every time. In other words, we have 26 choices of letters for the first letter. Once that is chosen we have 25 choices of letters for the second letter (any letter but the first). Similarly, we have 10 choices for the first digit, 9 for the second, and 8 for the third. We multiply all of the numbers together to get 26*25*10*9*8 = 468000.

This is essentially two permutations, both without repetition, so we multiply the results together. Read more about permutations without repetition here: https://www.mathsisfun.com/combinatorics/combinations-permutations.html

They also have some additional examples.