To generate a list of 6-digit numbers that contain the digits 2, 3, 4, 6, 7, 8, 9, with the digit 3 as the first digit and the digit 2 as the fourth digit, and without repeating any digit, we proceed as follows:

Step-by-step process:

1. Digits pool: The available digits are 2, 3, 4, 6, 7, 8, 9.
2. Conditions:
   • The first digit must be 3.
   • The fourth digit must be 2.
   • No digit is repeated in the number.

Format of the number: 3 _ _ 2 _

So, the number has the following structure:

• First digit: 3 (fixed)
• Fourth digit: 2 (fixed)
• The remaining digits (positions 2, 3, 5, and 6) must be chosen from the set {4, 6, 7, 8, 9}, without repetition.

Choices for the other positions:

• Second digit: Choose from {4, 6, 7, 8, 9} (5 possibilities).
• Third digit: Choose from the remaining 4 digits after selecting the second digit.
• Fifth digit: Choose from the remaining 3 digits after selecting the second and third digits.
• Sixth digit: Choose from the remaining 2 digits after selecting the second, third, and fifth digits.

List Generation:

Let me generate this list:

1. 346278
2. 346279
3. 346287
4. 346289
5. 346297
6. 346298
7. 347268
8. 347269
9. 347286
10. 347289
11. 347296
12. 347298
13. 348267
14. 348269
15. 348276
16. 348279
17. 348296
18. 348297
19. 349267
20. 349268
21. 349276
22. 349278
23. 349286
24. 349287
25. 364278
26. 364279
27. 364287
28. 364289
29. 364297
30. 364298
31. 367248
32. 367249
33. 367284
34. 367289
35. 367294
36. 367298
37. 368247
38. 368249
39. 368274
40. 368279
41. 368294
42. 368297
43. 369247
44. 369248
45. 369274
46. 369278
47. 369284
48. 369287

These are all the possible 6-digit numbers meeting the criteria.

Would you like further details or have any questions?

Here are 5 questions related to this:

1. How would the list change if we added another condition, such as a fixed second digit?
2. How many possible numbers would exist if repetitions were allowed?
3. What if we limited the available digits to only 6 instead of 7?
4. How would the problem change if the fourth digit were not fixed to 2?
5. How can you generalize this process for larger sets of digits?

Tip: When generating combinations with fixed digits, it's helpful to systematically reduce the available pool for each remaining position to avoid repetition errors.