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Part 1
How many different license plates are possible if each contains 2 letters​ (out of the​ alphabet's 26​ letters) followed by 4 digits​ (from 0 to​ 9)? How many of these license plates contain no repeated letters and no repeated​ digits?

To determine the number of different license plates possible, we need to consider the choices for each part of the license plate.

### Part 1: Total Number of License Plates

Each license plate consists of 2 letters followed by 4 digits.

1. **Letters**: There are 26 letters in the alphabet.
   - The first letter has 26 possible choices.
   - The second letter also has 26 possible choices.

2. **Digits**: Each digit can be from 0 to 9, which gives 10 possible choices for each digit.
   - The first digit has 10 possible choices.
   - The second digit has 10 possible choices.
   - The third digit has 10 possible choices.
   - The fourth digit has 10 possible choices.

Thus, the total number of license plates is calculated as:
\[
26 \times 26 \times 10 \times 10 \times 10 \times 10 = 26^2 \times 10^4
\]

Now, calculate the result:
\[
26^2 = 676
\]
\[
10^4 = 10000
\]
\[
676 \times 10000 = 6,760,000
\]

Therefore, the total number of different license plates possible is:
\[
6,760,000
\]

### Part 2: Number of License Plates with No Repeated Letters and No Repeated Digits

To find the number of license plates with no repeated letters and no repeated digits, we need to adjust our choices:

1. **Letters**: There are 26 letters in the alphabet.
   - The first letter has 26 possible choices.
   - The second letter has 25 possible choices (since it cannot be the same as the first letter).

2. **Digits**: Each digit can be from 0 to 9, but without repetition.
   - The first digit has 10 possible choices.
   - The second digit has 9 possible choices (since it cannot be the same as the first digit).
   - The third digit has 8 possible choices (since it cannot be the same as the first two digits).
   - The fourth digit has 7 possible choices (since it cannot be the same as the first three digits).

Thus, the total number of license plates without repeated letters and digits is calculated as:
\[
26 \times 25 \times 10 \times 9 \times 8 \times 7
\]

Now, calculate the result:
\[
26 \times 25 = 650
\]
\[
10 \times 9 = 90
\]
\[
90 \times 8 = 720
\]
\[
720 \times 7 = 5040
\]
\[
650 \times 5040 = 3,276,000
\]

Therefore, the number of license plates containing no repeated letters and no repeated digits is:
\[
3,276,000
\]

### Summary

1. **Total number of different license plates**: 6,760,000
2. **Number of license plates with no repeated letters and no repeated digits**: 3,276,000

Would you like more details on any part of the solution or have any other questions?

Here are 5 questions you might want to ask:
1. How are these calculations applicable in real-life scenarios?
2. Can you explain why the calculations for repeated and non-repeated characters are different?
3. How would the calculations change if there were more letters or digits allowed?
4. What if the license plate format was different, such as having more letters or fewer digits?
5. How do permutations and combinations differ in these types of problems?

**Tip:** When solving combinatorial problems, carefully distinguish between scenarios with and without repetition to apply the correct principles.

Learn how to calculate the total number of possible license plates with 2 letters and 4 digits, including those without repeated letters and digits. Understand the combinatorial principles involved in determining these counts. Suitable for high school level mathematics.

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