Using the digits 1,2,3,4,5 to create a 4 digit number sequence by combining two of the digits, at the beginning of the sequence and then listing the remaining numbers...example.... 12, 3, 4, 5. Or 54, 2, 3, 1. List all possibilities in order

To solve this problem, we will create all possible 4-digit sequences by first selecting two digits for the beginning of the sequence and then appending the remaining three digits. The total number of sequences will be the combinations of choosing two digits for the first part (there are 20 possibilities for that), and for each pair of chosen digits, there are 6 possible arrangements of the remaining digits.

Let me list all possibilities in order, starting with each pair:

### Possibilities:

1. **12**: 12, 3, 4, 5; 12, 3, 5, 4; 12, 4, 3, 5; 12, 4, 5, 3; 12, 5, 3, 4; 12, 5, 4, 3
2. **13**: 13, 2, 4, 5; 13, 2, 5, 4; 13, 4, 2, 5; 13, 4, 5, 2; 13, 5, 2, 4; 13, 5, 4, 2
3. **14**: 14, 2, 3, 5; 14, 2, 5, 3; 14, 3, 2, 5; 14, 3, 5, 2; 14, 5, 2, 3; 14, 5, 3, 2
4. **15**: 15, 2, 3, 4; 15, 2, 4, 3; 15, 3, 2, 4; 15, 3, 4, 2; 15, 4, 2, 3; 15, 4, 3, 2
5. **21**: 21, 3, 4, 5; 21, 3, 5, 4; 21, 4, 3, 5; 21, 4, 5, 3; 21, 5, 3, 4; 21, 5, 4, 3
6. **23**: 23, 1, 4, 5; 23, 1, 5, 4; 23, 4, 1, 5; 23, 4, 5, 1; 23, 5, 1, 4; 23, 5, 4, 1
7. **24**: 24, 1, 3, 5; 24, 1, 5, 3; 24, 3, 1, 5; 24, 3, 5, 1; 24, 5, 1, 3; 24, 5, 3, 1
8. **25**: 25, 1, 3, 4; 25, 1, 4, 3; 25, 3, 1, 4; 25, 3, 4, 1; 25, 4, 1, 3; 25, 4, 3, 1
9. **31**: 31, 2, 4, 5; 31, 2, 5, 4; 31, 4, 2, 5; 31, 4, 5, 2; 31, 5, 2, 4; 31, 5, 4, 2
10. **32**: 32, 1, 4, 5; 32, 1, 5, 4; 32, 4, 1, 5; 32, 4, 5, 1; 32, 5, 1, 4; 32, 5, 4, 1
11. **34**: 34, 1, 2, 5; 34, 1, 5, 2; 34, 2, 1, 5; 34, 2, 5, 1; 34, 5, 1, 2; 34, 5, 2, 1
12. **35**: 35, 1, 2, 4; 35, 1, 4, 2; 35, 2, 1, 4; 35, 2, 4, 1; 35, 4, 1, 2; 35, 4, 2, 1
13. **41**: 41, 2, 3, 5; 41, 2, 5, 3; 41, 3, 2, 5; 41, 3, 5, 2; 41, 5, 2, 3; 41, 5, 3, 2
14. **42**: 42, 1, 3, 5; 42, 1, 5, 3; 42, 3, 1, 5; 42, 3, 5, 1; 42, 5, 1, 3; 42, 5, 3, 1
15. **43**: 43, 1, 2, 5; 43, 1, 5, 2; 43, 2, 1, 5; 43, 2, 5, 1; 43, 5, 1, 2; 43, 5, 2, 1
16. **45**: 45, 1, 2, 3; 45, 1, 3, 2; 45, 2, 1, 3; 45, 2, 3, 1; 45, 3, 1, 2; 45, 3, 2, 1
17. **51**: 51, 2, 3, 4; 51, 2, 4, 3; 51, 3, 2, 4; 51, 3, 4, 2; 51, 4, 2, 3; 51, 4, 3, 2
18. **52**: 52, 1, 3, 4; 52, 1, 4, 3; 52, 3, 1, 4; 52, 3, 4, 1; 52, 4, 1, 3; 52, 4, 3, 1
19. **53**: 53, 1, 2, 4; 53, 1, 4, 2; 53, 2, 1, 4; 53, 2, 4, 1; 53, 4, 1, 2; 53, 4, 2, 1
20. **54**: 54, 1, 2, 3; 54, 1, 3, 2; 54, 2, 1, 3; 54, 2, 3, 1; 54, 3, 1, 2; 54, 3, 2, 1

This covers all the combinations of the form "XY, a, b, c" using the digits 1, 2, 3, 4, and 5.

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Would you like further clarification or detailed breakdowns for specific parts of this list?

### Related questions:
1. How would the pattern change if we used 6 digits instead of 5?
2. What is the mathematical formula behind generating such sequences?
3. How many unique combinations are there if repetition of digits is allowed?
4. Can you explain how to calculate the factorials used in these permutations?
5. What if we were restricted to using only odd or even digits?

**Tip:** For sequences involving multiple elements, understanding permutations and combinations helps simplify listing them!

Using the digits 1,2,3,4,5 to create a 4-digit number sequence by combining two of the digits at the beginning of the sequence and then listing the remaining numbers. Example: 12, 3, 4, 5. Or 54, 2, 3, 1. List all possibilities in order.

Explore how to create all possible 4-digit sequences using the digits 1, 2, 3, 4, 5 by selecting pairs of digits for the start and permuting the remaining digits. Includes detailed breakdown of the permutations and how factorials are applied. Suitable for students in Grades 8-10.

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