Subject identification numbers in a certain scientific research project consist of:

three digits followed by three letters and then three more digits.

Assume repetitions are not allowed within any of the three groups, but digits in the first group of three may occur also in the last group of three.

Determine the number of distinct identification numbers.

The number of distinct identification numbers is:  

To determine the number of distinct identification numbers given the conditions, let's break it down step by step:

### 1. Three Digits (First Group)
- There are 10 possible digits (0 through 9).
- Since repetitions are not allowed within this group:
  - The first digit can be any of the 10 digits.
  - The second digit can be any of the remaining 9 digits.
  - The third digit can be any of the remaining 8 digits.

The number of ways to choose these three digits is:
\[
10 \times 9 \times 8 = 720
\]

### 2. Three Letters (Middle Group)
- There are 26 possible letters (A through Z).
- Since repetitions are not allowed within this group:
  - The first letter can be any of the 26 letters.
  - The second letter can be any of the remaining 25 letters.
  - The third letter can be any of the remaining 24 letters.

The number of ways to choose these three letters is:
\[
26 \times 25 \times 24 = 15,600
\]

### 3. Three Digits (Last Group)
- Again, there are 10 possible digits (0 through 9).
- However, repetitions are not allowed within this group, but the digits in this group can be the same as the digits in the first group.
- The first digit can be any of the 10 digits.
- The second digit can be any of the remaining 9 digits.
- The third digit can be any of the remaining 8 digits.

The number of ways to choose these three digits is:
\[
10 \times 9 \times 8 = 720
\]

### 4. Total Number of Identification Numbers
- Since the three groups (first digits, letters, and last digits) are independent of each other, we multiply the number of ways to form each group to get the total number of distinct identification numbers:
\[
720 \times 15,600 \times 720
\]

Now, we calculate the total:
\[
720 \times 15,600 = 11,232,000
\]
\[
11,232,000 \times 720 = 8,087,040,000
\]

### Final Answer:
The number of distinct identification numbers is **8,087,040,000**.

---

Would you like further explanation or have any questions?

### Additional Practice Problems:
1. How many identification numbers are possible if all digits and letters can repeat?
2. How many identification numbers are possible if the digits in the first and last group cannot repeat across both groups?
3. How many identification numbers are possible if only the digits can repeat, but the letters cannot?
4. How many identification numbers are possible if all letters must be vowels?
5. How many identification numbers are possible if the first digit cannot be zero?
6. How many identification numbers are possible if all letters must be consonants?
7. How many identification numbers are possible if the first and last digits cannot be the same?
8. How many identification numbers are possible if at least one letter must be a vowel?

**Tip:** When dealing with combinations where order matters and repetition is restricted, always consider each group independently, and multiply the possibilities.

Explore how to calculate the number of distinct identification numbers with three digits, three letters, and three digits. Understand the permutations and combinations involved in forming these identification numbers under specified conditions. Suitable for advanced mathematics enthusiasts and students.

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