Question

*I am taking Math146: This question is a multipart question. I need to understand the reasoning because I am learning probability  

1. In the mid 1900’s, Connecticut license plates had six digits and the leading digit could not be 0. How many license plates were possible? Later in the century, Connecticut license plates were changed to have two letters followed by 4 digits. The leading letter could not be R, X, C, D, or Q. The second letter could not be Q. How many license plates were possible? At the end of the century, Connecticut license plates were again changed to have three numbers followed by three letters. The lead number could not be 0 and the lead letter could not be L. How many license plates were possible?

Answer 1

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Answer 2

Let's break down each part of the question and calculate the number of possible license plates step by step:

1. In the mid-1900s, Connecticut license plates had six digits, and the leading digit could not be 0. How many license plates were possible?

For the first digit, there are 9 choices (1 to 9) because it cannot be 0. For the second digit, there are 10 choices (0 to 9). For the third digit, there are 10 choices (0 to 9). For the fourth digit, there are 10 choices (0 to 9). For the fifth digit, there are 10 choices (0 to 9). For the sixth digit, there are 10 choices (0 to 9).

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

Total possible license plates = 9 (first digit) × 10^5 (rest of the digits) = 9 × 100,000 = 900,000

So, there are 900,000 possible license plates in this case.

2. Later in the century, Connecticut license plates were changed to have two letters followed by 4 digits. The leading letter could not be R, X, C, D, or Q. The second letter could not be Q. How many license plates were possible?

For the first letter, there are 24 choices (26 letters - 5 restricted letters). For the second letter, there are 25 choices (26 letters - 1 restricted letter). For the first digit, there are 10 choices (0 to 9). For the second digit, there are 10 choices (0 to 9). For the third digit, there are 10 choices (0 to 9). For the fourth digit, there are 10 choices (0 to 9).

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

Total possible license plates = 24 (first letter) × 25 (second letter) × 10^4 (digits) = 24 × 25 × 10,000 = 6,000,000

So, there are 6,000,000 possible license plates in this case.

3. At the end of the century, Connecticut license plates were changed to have three numbers followed by three letters. The lead number could not be 0, and the lead letter could not be L. How many license plates were possible?

For the first digit, there are 9 choices (1 to 9) because it cannot be 0. For the second digit, there are 10 choices (0 to 9). For the third digit, there are 10 choices (0 to 9). For the first letter, there are 25 choices (26 letters - 1 restricted letter). For the second letter, there are 26 choices (all letters are allowed). For the third letter, there are 26 choices (all letters are allowed).

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

Total possible license plates = 9 (first digit) × 10 (second digit) × 10 (third digit) × 25 (first letter) × 26^2 (remaining letters) = 9 × 10 × 10 × 25 × 676 = 1,710,000

So, there are 1,710,000 possible license plates in this case.