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JEE Advanced 2019 · Paper 2, Mathematics question 10

Five persons AA, BB, CC, DD and EE are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ______

Answer and worked solution

Official answer: 30

This counts proper colourings of a 5-cycle with 3 colours, given by
(k−1)n+(−1)n(k−1)=25−2=30.(k-1)^n+(-1)^n(k-1)=2^5-2=30.

Check by cases. Five people in a circle with 3 colours must use the pattern 2, 2, 1.

  • Choose the colour used once: 3 ways.
  • Choose who wears it: 5 ways.
  • The remaining four form a path that alternates the other two colours: 2 ways.

3×5×2=303\times5\times2=30.

Answer: 30

Topics:Distribution and arrangements
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