Please find every combination of 4 of these numbers (200,245,290,300,360,390,405,405,435,440,442,525,545,680) that add up to 1675.

To solve this problem, I will find all the unique combinations of four numbers from the given set that add up to 1675. The set of numbers is:

\[
\{200, 245, 290, 300, 360, 390, 405, 405, 435, 440, 442, 525, 545, 680\}
\]

I will calculate the valid combinations now.Here are the unique combinations of four numbers that add up to 1675:

1. \( 200 + 360 + 435 + 680 = 1675 \)
2. \( 200 + 390 + 405 + 680 = 1675 \) (appears twice due to duplicate 405)
3. \( 200 + 405 + 525 + 545 = 1675 \) (appears twice due to duplicate 405)
4. \( 245 + 360 + 390 + 680 = 1675 \)
5. \( 245 + 360 + 525 + 545 = 1675 \)
6. \( 290 + 300 + 405 + 680 = 1675 \) (appears twice due to duplicate 405)
7. \( 290 + 405 + 435 + 545 = 1675 \) (appears twice due to duplicate 405)
8. \( 300 + 390 + 440 + 545 = 1675 \)

Would you like details on any of these combinations or have further questions?

Here are some related questions:
1. How can we generalize this method for any target sum?
2. What is the probability of finding a specific sum in a randomly selected set of numbers?
3. How can we reduce redundancy when there are repeated numbers?
4. How does this approach scale with larger sets of numbers?
5. How could dynamic programming help optimize the process?

**Tip**: When solving combination problems, be mindful of repeated elements, as they can lead to duplicate solutions.

Explore how to find every combination of four numbers from the set (200, 245, 290, 300, 360, 390, 405, 405, 435, 440, 442, 525, 545, 680) that sum up to 1675. This problem involves concepts of combinatorics and number theory, suitable for students in Grades 9-12.

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