Question
For many years, the state of California used 3 letters followed by 3 digits on its automobile license plates. (a) How many different license plates are possible with this arrangement? (b) When the state ran out of new numbers, the order was reversed to 3 digits followed by 3 letters. How many new license plate numbers were then possible? (c) By 1980, the numbers described in part (b) were also used up. The state then issued plates with 1 digit followed by 3 letters and then 3 digits. How many new license plate numbers will this provide?
Solution
VerifiedStep 1
1 of 4a.
There are 26 letters to choose from. There are 10 digits to choose from.
We have the following choices:

The 1st letter can be chosen in 26 ways.

The 2nd letter can be chosen in 26 ways.

The 3rd letter can be chosen in 26 ways.

The 1st digit can be chosen in 10 ways.

The 2nd digit can be chosen in 10 ways.

The 3rd digit can be chosen in 10 ways.
By the multiplication principle, total = $26^{3}\times 10^{3}=17,576,000$