How Many Possible License Plates?

A tree diagram can be handy for counting things like possible license plate arrangements.

Problem A license plate starts with three letters, C, N, and B, in an unknown order. After the three letters are three numbers, 7, 3, and 4, in an unknown order. How many license plates are there with three letters followed by three numbers (chosen from the one listed above)? Assume letters and numbers are not repeated.

Solution The picture above shows one possible license plate, let’s look at other possibilities. Start by making a tree for the three letters at the beginning of the license plate.

Based on this tree, there are 6 ways to rearrange the letters C, B, and N, without repeating letters.

We can make a similar tree for the numbers.

There are 6 ways to rearrange the numbers. So if there are 6 ways to rearrange the letters and each of those rearrangements have 6 ways to rearrange the numbers, there are 6 × 6 = 36 different license plates.