What is the number of selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction?
- A3
- B11
- C12Correct
- D66
Explanation
In a circle of 12 things, a selection of 10 consecutive things is uniquely determined by its starting point. Since there are 12 distinct things, any of these 12 things can serve as the starting point for a sequence of 10 consecutive things taken in the clockwise direction.
For example:
- Starting with thing 1, we select (1, 2, 3, 4, 5, 6, 7, 8, 9, 10).
- Starting with thing 2, we select (2, 3, 4, 5, 6, 7, 8, 9, 10, 11).
- Starting with thing 3, we select (3, 4, 5, 6, 7, 8, 9, 10, 11, 12). ...and so on, until...
- Starting with thing 12, we select (12, 1, 2, 3, 4, 5, 6, 7, 8, 9).
Each of these 12 selections is distinct and satisfies the condition of being 10 consecutive things in a clockwise direction.
Analyzing the options: A) 3: This is incorrect. B) 11: This is incorrect. C) 12: This is the correct answer, as there are 12 possible starting points for the consecutive sequence. D) 66: This would be the number of ways to choose any 10 things out of 12 (calculated as C(12,10) or C(12,2)), without the crucial "consecutive" constraint.
The final answer is C) 12.

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