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To determine the next combination based on the given sets of numbers, let's analyze the sequences:

1. **First row:** 12, 15, 20, 3, 13, 34
2. **Second row:** 24, 12, 17, 33, 23, 40
3. **Third row:** 39, 33, 20, 7, 4, 12

### Step 1: Look for patterns in each column:
- **Column 1:** 12, 24, 39
   - The difference: 24 - 12 = 12, 39 - 24 = 15. The difference is increasing by 3.
   - The next number would be \( 39 + 18 = 57 \).
  
- **Column 2:** 15, 12, 33
   - The difference: 12 - 15 = -3, 33 - 12 = 21. There is no consistent difference, but alternating between a negative and a large positive change suggests some irregularity.
   - To follow this alternating pattern, the next number could involve a smaller positive change (e.g., \( 33 + (-6) = 27 \)).

- **Column 3:** 20, 17, 20
   - The difference: 17 - 20 = -3, 20 - 17 = 3. It alternates between -3 and +3.
   - The next number would be \( 20 - 3 = 17 \).
  
- **Column 4:** 3, 33, 7
   - The difference: 33 - 3 = 30, 7 - 33 = -26. The differences alternate drastically.
   - The next number would be \( 7 + 30 = 37 \).

- **Column 5:** 13, 23, 4
   - The difference: 23 - 13 = 10, 4 - 23 = -19. The differences fluctuate significantly.
   - The next number might follow with an increase, \( 4 + 10 = 14 \).

- **Column 6:** 34, 40, 12
   - The difference: 40 - 34 = 6, 12 - 40 = -28. This alternates between positive and negative differences.
   - The next number would be \( 12 + 6 = 18 \).

### Step 2: Next combination:
Based on the above analysis, the next combination would likely be:

**57, 27, 17, 37, 14, 18**

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Would you like more details on this process or have any further questions?

Here are some related questions for practice:

1. How can you identify patterns in number sequences?
2. What methods help in predicting the next number in alternating sequences?
3. How do you handle irregular number differences in sequences?
4. Can geometric progressions also be applied to such problems?
5. What other number sequences do you commonly encounter in mathematical problems?

**Tip:** When working with sequences, always check for alternating or repeating patterns in both the differences and ratios between numbers.

What is the next combination in the sequence 12 15 20 3 13 34, 24 12 17 33 23 40, 39 33 20 7 4 12?

Learn how to predict the next number combination in a given sequence using pattern recognition. This problem explores number sequences with alternating differences and irregular patterns, suitable for students in Grades 9-12.

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