Permutations and Combinations
Derangements
JEE Advanced 2019 Paper 2
Grade 11
Question:
Five persons A, B, C, D, E are seated in a fixed circular arrangement. Each is given a hat of one of three colours: red, blue, or green. Let $P$ be the number of arrangements such that any two adjacent persons have different coloured hats. Then $P$ equals _____.
Step-by-Step Solution
Key Concept: This is the chromatic polynomial of the cycle $C_5$ evaluated at $k = 3$: $P(C_5, k) = (k - 1)^5 + (-(k-1)) = 2^5+(-2)(-1)^5 = (k-1)^5+(-1)^5(k-1)$. With $k = 3$: $(3-1)^5+(-1)^5(3-1) = 32 - 2 = 30$.
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Correct Answer: 30