Suppose each license plate in a certain state has three letters followed by three digits. The letters H and L and the digits 1,2, and 9  are not used. So, there are  24 letters and  7 digits that are used. Assume that the letters and digits can be repeated. How many license plates can be generated using this format?
 


To determine the total number of possible license plates, we can break the problem into two parts: calculating the number of choices for letters and the number of choices for digits.

### Step 1: Finding the Number of Choices for Letters
Each license plate has 3 letters, and the letters **H** and **L** are not used. That means we are left with \(26 - 2 = 24\) letters.

Since letters can be repeated, the total number of possibilities for the 3-letter combination is:
\[
24 \times 24 \times 24 = 24^3
\]

### Step 2: Finding the Number of Choices for Digits
Each license plate also has 3 digits, and the digits **1**, **2**, and **9** are not used. That leaves \(10 - 3 = 7\) digits.

With repetition allowed, the total number of possibilities for the 3-digit combination is:
\[
7 \times 7 \times 7 = 7^3
\]

### Step 3: Combining Letters and Digits
To find the total number of license plates, multiply the number of possibilities for the letters and digits:
\[
24^3 \times 7^3
\]

Calculating each value separately:
\[
24^3 = 24 \times 24 \times 24 = 13,824
\]
\[
7^3 = 7 \times 7 \times 7 = 343
\]

Now, multiply these two results:
\[
13,824 \times 343 = 4,742,832
\]

### Final Answer
The total number of license plates that can be generated is:
\[
\boxed{4,742,832}
\]

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Would you like more details, or have any questions?

### Related Questions
1. How would the solution change if repetition of letters was not allowed?
2. What if each plate had 4 letters followed by 2 digits instead?
3. How many license plates can be formed if only vowels were excluded from the letters?
4. What if the format was 2 letters followed by 4 digits?
5. What is the probability of generating a plate with the same letter repeated three times?

### Tip:
When dealing with repetition problems, count the number of possibilities for each position and multiply them together.

Suppose each license plate in a certain state has three letters followed by three digits. The letters H and L and the digits 1, 2, and 9 are not used. So, there are 24 letters and 7 digits that are used. Assume that the letters and digits can be repeated. How many license plates can be generated using this format?

Learn how to calculate the total number of license plates that can be generated using 24 letters (excluding H and L) and 7 digits (excluding 1, 2, and 9). The problem involves key combinatorial concepts like permutations with repetition and the fundamental counting principle. Suitable for students in Grades 9-12.

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