(A) Let the number of ways to color $n$ vertices of a cycle graph with $k$ colors such that no two adjacent vertices have the same color be $P(n, k)$.

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(A) Let the number of ways to color $n$ vertices of a cycle graph with $k$ colors such that no two adjacent vertices have the same color be $P(n, k)$.This is given by the formula $P(n, k) = (k-1)^n + (-1)^n(k-1)$.Here,$n = 5$ (number of persons) and $k = 3$ (number of colors).Substituting the values,we get:$P(5, 3) = (3-1)^5 + (-1)^5(3-1)$$P(5, 3) = 2^5 - 2$$P(5, 3) = 32 - 2 = 30$.Thus,the total number of ways is $30$.

Class 11 Mathematics Permutations and Combinations Circular permutations

Medium

Five persons $A, B, C, D$ and $E$ 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

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$30$
B
$40$
C
$50$
D
$60$

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Five persons $A, B, C, D$ and $E$ 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

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