Find every way that 4 of these numbers  (200,245, 290, 300, 300 again, 309, 310, 314, 318, 325, 350, 350 again, 360, 360 again, 385, 390, 405, 405 again, 435, 440, 442, 460, 490, 495, 525, 535, 545, 670, 680, 680 again, and 680 again) add up to 1864.

To find all combinations of four numbers from the given list that add up to 1864, we can break this down into a step-by-step computational process.

Given the numbers:
- 200, 245, 290, 300, 300, 309, 310, 314, 318, 325, 350, 350, 360, 360, 385, 390, 405, 405, 435, 440, 442, 460, 490, 495, 525, 535, 545, 670, 680, 680, 680

The task is to find all distinct combinations of four numbers whose sum equals exactly 1864. To do this manually would be quite exhaustive, so I'll outline the computational approach below:

### Approach:

1. **Generate all possible sets of four numbers** from the list, making sure to account for repeated numbers like 300, 350, 360, 405, and 680 which appear multiple times.
  
2. **Sum each combination** and filter out the ones that add up to exactly 1864.

### Solution via Computation:

I'll calculate the results using Python to ensure we find all correct combinations. Let me do this computation.Here are the valid combinations of four numbers from the given list that add up to 1864:

1. (200, 314, 670, 680) – appears 3 times due to repeated numbers.
2. (309, 350, 525, 680) – appears 6 times due to repeated numbers.
3. (309, 350, 535, 670) – appears 2 times.
4. (309, 360, 525, 670) – appears 2 times.
5. (309, 385, 490, 680) – appears 3 times.
6. (309, 390, 495, 670)
7. (309, 435, 440, 680) – appears 3 times.
8. (309, 495, 525, 535)
9. (314, 325, 545, 680) – appears 3 times.
10. (314, 385, 495, 670)
11. (314, 390, 490, 670)
12. (314, 490, 525, 535)

These combinations are valid solutions to the problem.

Would you like further details on any of these combinations, or another clarification?

### Related questions:
1. What happens if the list contains more repeated numbers?
2. How can we adapt this method to find combinations summing to a different total?
3. Can this technique be applied to larger sets of numbers?
4. How would the result change if only unique combinations were considered?
5. How can such a problem be solved more efficiently for very large lists?

**Tip:** Always double-check the presence of repeated numbers when solving combination problems to ensure no valid combination is missed.

Find every way that 4 of these numbers (200, 245, 290, 300, 300 again, 309, 310, 314, 318, 325, 350, 350 again, 360, 360 again, 385, 390, 405, 405 again, 435, 440, 442, 460, 490, 495, 525, 535, 545, 670, 680, 680 again, and 680 again) add up to 1864.

Discover how to find all possible combinations of four numbers from a list that add up to 1864, including handling repeated elements. This problem explores combinatorics and addition with step-by-step solutions. Ideal for students in Grades 9-12.

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