Question

A state's license plates consist of four letters followed by three numerals, and 241 letter arrangements are not allowed. How many plates can the state issue?

Answer 1

There are 26 alphabet characters (A-Z) available to choose from and 10 numbers (0-9) to choose. Normally, lower-case letters aren't a possibility, some let's exclude them.

For the 1st space in the licence plate we can choose from any of the 26 characters so there are 26 choices
For the second place in the license plate, same thing, 26 characters to choose from for that space
So for these first two spaces, there are 26*26 possibilities.
We go on to the 3rd and 4th space, same idea.
By the 4th space, we have 26*26*26*26 possible combinations
Continuing on to the 5th space, we can have any of 10 numbers, so ten possibilities for this space
Similarly for the 6th and 7th space
So for the last three spaces, we have 10*10*10 ways to arrange 3 spaces that can be numbers
Combining these possibilities with the 1st four spaces, we have 26*26*26*26*10*10*10 possibilities total
However, since 241 possibilities will be excluded (probably for good reason), we need to subtract if from the previous number

That's it!

Any questions?