CSAT Solved Papers/ 2023/Q48

2023 CSAT — Q48

Quant Counting & combinatorics 2.5 marks Easy

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 Answer
D 66

Worked rationale

A block of ” $10$ consecutive things taken clockwise” from a circle of $12$ is completely determined by its starting position. In a circle of $n$ items, the number of consecutive clockwise arcs of any fixed length $k$ (with $1 \le k \le n$ ) equals the number of starting points:

n = 12.

Each of the $12$ positions begins a distinct clockwise run of $10$ , so there are $12$ such selections.

Answer: (c) 12.

Why the other options miss

A

solved the wrong shape: computes $12 - 10 + 1 = 3$ , the linear (row) count of consecutive blocks, ignoring that a circle has no endpoints.
B

off by one: treats it as a near-circular count $n - 1$ , dropping one rotational start.
D

wrong formula: uses $\binom{12}{2} = 66$ (choosing the $2$ excluded items as if unordered/non-adjacent), forgetting the excluded pair must itself be adjacent for the kept block to be consecutive.

Specialist insight

The circle is the trap. In a line, consecutive blocks of length $k$ number $n-k+1$ ; in a circle, they number $n$ (every position is a valid start, wrapping around). Equivalently, choosing $10$ consecutive to keep is the same as choosing the $2$ consecutive to drop, and there are exactly $12$ adjacent pairs in a $12$ -cycle. Either view gives $12$ . The decoy $3$ is the linear answer; $66$ ignores adjacency.

The trap, in one line

In a circle of $12$ , consecutive arcs of length $10$ number $12$ (one per start) — not $12-10+1 = 3$ (that's the line) $\Rightarrow$ (c).

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