Answer:
47,174,400 possible number plates
See important note at the bottom of the explanation
Explanation:
The easiest way to handle a problem of this type is to note the following
There are 9 digits 1-9 that we can use for the first four characters of the license plate
- This means that for the first digit there are all 9 possibilities
- For the second digit, since there is no repetition, there exists only 8 possibilities
- For the third spot, there are 7 possibilities
- For the 4th spot there are 6 possibilities
So the number of ways to arrange the first 4 digits without repetition is
9 x 8 x 7 x 6 = 3024 possible ways
For the letters, assuming that we are using the English alphabet, the 5th spot where the first letter after the numbers is can be done in 26 possible ways since there are 26 alphabets.
For the second letter that will leave us with 25 possibilities
For the third and final spot there are 24 possibilities
This means that, without repetition, the letters can be arranged in 26 x 25 x 24 ways = 15,600 ways
Together we can arrange the digits and letters in 3024 x 15600 ways which works out to
3024 x 15600 = 47,174,400 ways or possible number plates
Or we could simply write this in one step as
9 x 8 x 7 x 6 x 26 x 25 x 24 = 47,174,400 ways or possible number plates
Note
Even though there are 10 digits 0 - 9, the question clearly states that we should choose from digits 1 through 9 only.
If we are allowed to use all 10 digits the first 4 digits can be put together in 10 x 9 x 8 x 7 = 78,624,000 ways
You should clarify this with your teacher.