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# How many license plates consist of $3$ letters, followed by $2$ even digits, followed by $2$ odd digits?

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How many license plates consist of $3$ letters, followed by $2$ even digits, followed by $2$ odd digits?

Jul 15, 2023

#1
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I assume that both the letters and the digits can be repeated. If so, then:

Example: AAA 22 33

You have 26 choices for the first "A", 26 choices for the 2nd "A", and 26 choices for the 3rd "A"

That is: 26 x 26 x 26

For EVEN digits, you have 5 choices: 0, 2, 4, 6, 8

For the first digit, you have 5 choices, and for the 2nd digit you have 5 choices.

That is: 5 x 5

For ODD digits, you have 5 choices: 1, 3, 5, 7, 9

For the first digit, you have 5 choices, and for the 2nd digit you have 5 choices.

That is: 5 x 5

We put all together and we have: 26 x 26 x 26 x 5 x 5 x 5 x 5 ==10,985,000 license plates possible.

Jul 15, 2023
#2
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Hello,
To determine the number of license plates consisting of 3 letters, followed by 2 even digits, and ending with 2 odd digits, we need to consider the available choices for each position.

Letters: There are 26 letters in the English alphabet. Since repetition is allowed, there are $26^3$ ways to choose 3 letters.

Even Digits: The even digits available are 0, 2, 4, 6, and 8. Since repetition is allowed, there are $5^2$ ways to choose 2 even digits.

Odd Digits: The odd digits available are 1, 3, 5, 7, and 9. Since repetition is allowed, there are $5^2$ ways to choose 2 odd digits.

To find the total number of license plates, we multiply the number of choices for each position:

$26^3 \times 5^2 \times 5^2 = 26^3 \times 5^4 = 175,760,000.$

Therefore, there are 175,760,000 license plates that consist of 3 letters, followed by 2 even digits, and ending with 2 odd digits.

Jul 17, 2023