5. PA license plates have 3 letters followed by 4 numbers. a. If the same letter or number can be repeated, how many can be made?
b. If the same letter CANNOT be repeated, how many can be made?

A. 175,760,000

B. 156,000,000

Step-by-step explanation:

For part A:

There are 26 letters in the alphabet.  Therefore, if there are 3 spots to use 26 letters, that is represented as .  As for numbers, there are 10 digits from 0-9.  So if there are 4 spots for numbers, the total possible number of sequences is .  So you simply multiply and to get 175,760,000 possible combinations.

For part B:

If the same letter can't be repeated, instead of using an exponent, we have to use a factorial, but only for the 3 digits required in the license plate.  These are called permutations and can be expressed as , which is the same as saying 26 x 25 x 24.  Since there is no change to number of numbers, that remains the same.