If a license plate has three letters followed by three digits, with no repetitions allowed and no use of either the letter o or the digit 0, how many different license plates are possible?

The license plates will look like this:

letter1 letter2 letter3 digit1 digit2 digit3

A sample license plate looks like this: ACT983

We need to see the numbers of letters and digits that can be used in each place, and multiply all numbers together.

You cannot use the letter O. Since there are 26 letters in the alphabet, by removing the letter O, you now have 25 letters left. The position "letter1" can have one of 25 letters.

25

For letter2, you cannot use the letter O, and you cannot use the letter that was already used, so now there are 24 letters left. Position letter2 can be filled with on of the remaining 24 different letters.

25 * 24

Letter3 can have 23 choices of letters.

25 * 24 * 23

Now we do a similar thing with the digits. Digit1 cannot be zero, but it can be any digit from 1 to 9, so digit1 has 9 choices. Since no repetitions are allowed, digit2 has 8 choices of digits, and then digit3 has 7 choices of digits.

The number of different license plates available is the product:

25 * 24 * 23 * 9 * 8 * 7

25 * 24 * 23 * 9 * 8 * 7 = 6,955,200

Answer: 6,955,200