how may possibilities are there for each of the following license plate schemes

Question

a. three letters followed by three digits

b. one digit followed by three letters followed by three digits

Answer 1

a. This question involves permutations. It doesn't specify whether letters or digits can be repeated, but I will assume that they can be. the alphabet has 26 letters, so we have 26 choices for each of the first three places on the license plate. There are 10 digits, including zero, so we have 10 choices for each of the last three positions. We set up a multiplication pattern like this:

26 X 26 X 26 X 10  X 10 X 10   =  17,576,000

Note: if letters and/or digits cannot be repeated, you would have to subtract one for each new position. If neither can be repeated, it would be:

26 X 25 X 24 X 10 X 9 X 8  =  11,232,000

 

b. This involves the same process as a). If both letters and digits can be repeated, the setup is:

10 X 26 X 26 X 26 X 10 X 10 X 10 =  175,760,000

If neither can be repeated, and if you cannot start a license number with a 0, it would be this:

9 X 26 X 25 X 24 X 9 X 8 X 7  =  70,761,600

You can adjust for any other restrictions by changing the appropriate numbers. For example, if the first letter must be a vowel, that letter space would be a 5 instead of 26 because the alphabet has five vowels, not countiing y.