Introduction:
To find the sum of all the numbers that can be formed using the digits 1, 3, 5, and 7 without any repetition, we need to analyze the possible combinations and add them up. Let's break down the problem step by step.

Step 1: Finding all possible combinations:
To form different numbers, we need to use all the given digits without repetition. This can be done by taking each digit as the first digit and combining it with the remaining three digits in a specific order.

The total number of combinations can be calculated using the formula for permutations. In this case, since we are taking all the digits at once, the formula is:

Number of combinations = 4P4 = 4!

Step 2: Identifying the combinations:
Let's list down all the combinations that can be formed using the given digits:

1. 1357
2. 1375
3. 1537
4. 1573
5. 1735
6. 1753
7. 3157
8. 3175
9. 3517
10. 3571
11. 3715
12. 3751
13. 5137
14. 5173
15. 5317
16. 5371
17. 5713
18. 5731
19. 7135
20. 7153
21. 7315
22. 7351
23. 7513
24. 7531

Step 3: Calculating the sum:
To find the sum of all these numbers, we can add them up individually. Let's calculate the sum:

Sum = 1357 + 1375 + 1537 + 1573 + 1735 + 1753 + 3157 + 3175 + 3517 + 3571 + 3715 + 3751 + 5137 + 5173 + 5317 + 5371 + 5713 + 5731 + 7135 + 7153 + 7315 + 7351 + 7513 + 7531

Sum = 54104

Conclusion:
The sum of all the numbers that can be formed using the digits 1, 3, 5, and 7 without any repetition is 54104.