Round table seating with constraints

CAT 1997 Slot 1 · QA · Medium · Permutations and Combinations

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In how many ways can eight directors, the vice chairman, and chairman of a firm be seated at a round table, if the chairman has to sit between the vice chairman and a director?

  1. A.

    9! × 2

  2. B.

    2 × 8!

  3. C.

    2 × 7!

  4. D.

    None of these

Answer

B

Explanation

There are 10 people in total: 8 directors, 1 vice chairman (VC), and 1 chairman (C). The chairman must sit between the VC and a director. Thus, the sequence VC - C - Director must form a block of 3, or C is fixed between VC and one director.

More simply, treat VC and C as a combined unit where C is next to VC. Since C must be between VC and a Director, C sits adjacent to VC on one side and a director on the other.

First, arrange the 8 directors around the circular table in (8−1)!=7!(8-1)! = 7! ways, or position the 8 directors around the table. There are 8 gaps between the 8 directors where the pair (VC, C) can be placed. In any chosen gap, C can be placed next to one director and VC next to C in 2 relative orientations (VC-C or C-VC).

Total ways = 8!×2=2×8!8! \times 2 = 2 \times 8!.

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