What is the number of selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction ?

(a) 3
(b) 11
(c) 12 ✓ UPSC's answer
(d) 66

Why the answer is (c)

• The problem asks for the number of ways to select 10 consecutive items from 12 items arranged in a circle, moving in a clockwise direction.

• In a circular arrangement of $n$ items, the number of ways to select $k$ consecutive items is equal to $n$, provided $k < n$.

• This is because each of the 12 items can serve as the starting point of the sequence of 10 consecutive items.

• For example, starting at item 1, the selection is {1, 2, ..., 10}; starting at item 2, it is {2, 3, ..., 11}; and so on.

• Since there are 12 distinct starting positions, there are exactly 12 such selections.

• Therefore, the correct answer is 12.

Why the other options are wrong

(a) 3: Option 3 is incorrect because it does not account for the 12 distinct starting positions available in the circle.
(b) 11: Option 11 is incorrect because it undercounts the total number of starting positions, which is equal to the total number of items (12).
(d) 66: Option 66 is incorrect because it represents the combination formula $\binom{12}{2}$ or similar non-consecutive selection logic, not consecutive selections in a circle.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.