How many pattern passwords can be made using at least 3 out of 9 dots?

To calculate the number of patterns you can make using at least 3 out of 9 dots, we can break it down into cases:

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!