Question

Suppose that in a state, license plates have exactly three letters and three numbers. The numbers and letters can appear anywhere, and repetitions are allowed.

What is the probability that a randomly selected vehicle will have a license plate with no repetitions of any characters and which include the number 6 but does not include the letter A?

--

S → all possible plates

A → event plate mets given criteria

I know that |S| = 26^3 * 10^3 * (6 chose 3). Why is the 6 chose 3 necessary?
I also know that |A| = (6 chose 3) * (25 chose 3) * 3! * (9 chose 2) * 3! . Where do the 3!s come from (why are they necessary?) and why does this also contain 6 chose 3?

Answer 1

Explanation:

Let's break down the problem step by step:

1. The total number of possible license plates:

- There are 26 letters (A-Z) and 10 digits (0-9).

- The license plate has exactly 3 letters and 3 numbers.

- Repetitions are allowed, so there are 26 choices for each letter and 10 choices for each number.

- The total number of possible plates is 26^3 * 10^3.

2. The probability that a randomly selected vehicle will have a license plate with no repetitions of any characters and includes the number 6 but does not include the letter A:

For a plate to meet these criteria, you have several conditions:

- The license plate includes the number 6.

- The license plate does not include the letter A.

- There are no repetitions of any characters (letters or numbers).

Now, let's calculate |A|, the number of plates that meet these criteria:

- Choose 3 positions out of 6 for the numbers (this is the "6 choose 3" part). This determines where the numbers 6 will go on the plate.

- Choose 3 different letters from the 25 remaining letters (since A is not allowed).

- There are 3! ways to arrange the numbers 6 in the chosen positions.

- There are 3! ways to arrange the selected letters in the remaining positions.

- Choose 2 different numbers from the 9 remaining numbers (excluding 6) to fill the remaining two number positions.

So, |A| = (6 choose 3) * (25 choose 3) * 3! * 3! * (9 choose 2).

Now, let's address your specific questions:

- **Why is "6 choose 3" necessary?**: The "6 choose 3" part represents the ways to select 3 positions out of 6 for the numbers 6. It's necessary because you need to determine where the number 6 will appear on the license plate.

- **Why are there 3!s in the formula?**: The 3!s account for the different arrangements of the numbers 6 and the selected letters. Since there are no repetitions allowed, you need to consider the number of ways to arrange the characters within the selected positions.

In summary, the "6 choose 3" and the 3! terms are necessary to account for the different ways you can select positions and arrange characters on the license plate while meeting the specified criteria.