Question

If a license plate consists of four digits, how many different licenses could be created having at least one digit repeated?

Answer 1

4,960

Explanation:

I'm going to assume that the constraint "have at least one digit repeating" means that the presence of the same digit in 2 or more places in the 4 places allowed will constitute repetition.

To find the number of plates that has at least one digit repeat, we can find the number of plates we can make in total, then calculate the number of plates that have no repeating digits and subtract that from the total and that will give us the number that have at least one repeating digit. Mathematically, I can say:

`"total number of plates" - "number of plates with no repeating digits"="number of plates with at least one repeating digit"`

So how many plates can be made in total?

If a plate consists of 4 digits and there are 10 digits we can pick from, the number of plates we can make can be found this way:

The first digit of the plate has 10 choices = 10
So does the second digit = 10
And the third = 10
And the fourth = 10

So we can make `10xx10xx10xx10=10,000` plates

How many will have no repeats?

Here, the first digit can be anything, and so we have 10 choices = 10

The second digit can be anything except the first digit, so that's 9 choices = 9

The third digit can be anything except for the first 2 = 8

And the last digit can be anything except the first 3 = 7

That gives us:

`10xx9xx8xx7=5040`

And so the number of plates that have at least 1 repeat is:

`10,000-5,040=4,960`