Non-Adjacent Pairs Singing in Circle

CAT 2004 Slot 1 · QA · Medium · Permutations and Combinations

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NN persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 2828 minutes, what is NN?

  1. A.

    55

  2. B.

    77

  3. C.

    99

  4. D.

    None of the above

Answer

B

Explanation

Total pairs of NN persons is (N2)=N(N−1)2\binom{N}{2} = \frac{N(N-1)}{2}. The number of adjacent pairs is NN. Thus, number of non-adjacent pairs =N(N−1)2−N=N(N−3)2= \frac{N(N-1)}{2} - N = \frac{N(N-3)}{2}. Each song takes 2 minutes, so total time is: 2×N(N−3)2=N(N−3)=282 \times \frac{N(N-3)}{2} = N(N-3) = 28 N2−3N−28=0  ⟹  (N−7)(N+4)=0N^2 - 3N - 28 = 0 \implies (N - 7)(N + 4) = 0 Since N>0N > 0, N=7N = 7.

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