A Califorina license plate consists of three letters followed by three digits.

Question

A) How many license plates are possible?

B) How many license plates are possible if the letters are in alphabetical order and the digits in increasing order?

Answer 1

This is a very complicated question... It is going to be a very high number. Let's take a look and see if we can get close.

 

It is important to note here that there are 26 letters in the alphabet (A-Z), and 10 possible numbers (0-9) that can be used.

 

Lets start by writing out a couple of license plates:

 

AAA-111

AAB-111

...

AAZ-111

 

Notice that I only changed 1 letter. This indicates that there are 26 possible license plates that begin with AA and end with 111. Now let's change a different letter.

 

ABA-111

ABB-111

...

ABZ-111

 

Again, only 1 letter was changed. Now there are 26 more possibilities beginning with AB and ending with 111. This is a pattern that will continue all of the way down to AZ_-111. To find all of the possibilities That begin with the letter A and end with the numbers 111, we multiply 26 by 26.

 

26*26 = 676

 

Now, for each of those 676 possibilities, they can be followed by any combination of numbers. Let's write out a couple of more license plates.

 

AAA-000

AAA-001

...

AAA-009

 

So for each license plate beginning with AAA-00, there are 10 possible choices as to what the last number is. Now let us write out a few more plates:

 

AAA-010

AAA-011

...

AAA-019

 

Again, there are 10 possibilities. So we multiply 10 and 10 to get 100. But we also need to consider that the first number is not 0. There are another 10 possibilities there. So we multiply 100 by 10 to get 1000.

 

So now for each license plate beginning with A we have:

676 letter combinations * 1000 number combinations = 676,000

 

We also have 26 possible letters that the license plate can start with, so:

 

676,000 * 26 = 17,576,000 possible license plates.

 

Now, generally license plates do not allow the letters I and O because they look like 1's and 0's. If that is the case with this problem then simply replace every 26 with a 24 and re-do the math.

 

**Note, this does allow for the use of the same numbers multiple times.