Case 1: 3 dots are selected
- There are 9 ways to choose the first dot, 8 ways to choose the second dot, and 7 ways to choose the third dot.
- So, there are 9 * 8 * 7 = 504 patterns for this case.
Case 2: 4 dots are selected
- There are 9 ways to choose the first dot, 8 ways to choose the second dot, 7 ways to choose the third dot, and 6 ways to choose the fourth dot.
- So, there are 9 * 8 * 7 * 6 = 3024 patterns for this case.
Case 3: 5 dots are selected
- There are 9 ways to choose the first dot, 8 ways to choose the second dot, 7 ways to choose the third dot, 6 ways to choose the fourth dot, and 5 ways to choose the fifth dot.
- So, there are 9 * 8 * 7 * 6 * 5 = 15120 patterns for this case.
Finally, add up the patterns from all the cases:
504 + 3024 + 15120 = 18648 patterns.
Therefore, there are 18,648 different patterns that can be made using at least 3 out of 9 dots.
Let's include those cases as well.
Case 4: 6 dots are selected
- There are 9 ways to choose the first dot, 8 ways to choose the second dot, 7 ways to choose the third dot, 6 ways to choose the fourth dot, 5 ways to choose the fifth dot, and 4 ways to choose the sixth dot.
- So, there are 9 * 8 * 7 * 6 * 5 * 4 = 15120 patterns for this case.
Case 5: 7 dots are selected
- There are 9 ways to choose the first dot, 8 ways to choose the second dot, 7 ways to choose the third dot, 6 ways to choose the fourth dot, 5 ways to choose the fifth dot, 4 ways to choose the sixth dot, and 3 ways to choose the seventh dot.
- So, there are 9 * 8 * 7 * 6 * 5 * 4 * 3 = 30240 patterns for this case.
Case 6: 8 dots are selected
- There are 9 ways to choose the first dot, 8 ways to choose the second dot, 7 ways to choose the third dot, 6 ways to choose the fourth dot, 5 ways to choose the fifth dot, 4 ways to choose the sixth dot, 3 ways to choose the seventh dot, and 2 ways to choose the eighth dot.
- So, there are 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 = 60480 patterns for this case.
Case 7: 9 dots are selected
- There are 9 ways to choose each of the nine dots.
- So, there are 9^9 = 387420489 patterns for this case.
Finally, add up the patterns from all the cases:
15120 + 30240 + 60480 + 387420489 = 387516329 patterns.
Therefore, there are 387,516,329 different patterns that can be made using at least 3 out of 9 dots, including all 9 dots. Thank you for bringing this to my attention!
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