(A) Let the number of ways to colour a circle of $n$ vertices with $k$ colours such that no two adjacent vertices have the same colour be $P_n(k)$.For

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(A) Let the number of ways to colour a circle of $n$ vertices with $k$ colours such that no two adjacent vertices have the same colour be $P_n(k)$.For a circular arrangement,the formula is given by $P_n(k) = (k-1)^n + (-1)^n(k-1)$.Here,$n = 5$ (number of persons) and $k = 3$ (number of colours).Substituting the values,we get $P_5(3) = (3-1)^5 + (-1)^5(3-1)$.$P_5(3) = 2^5 - 1(2) = 32 - 2 = 30$.Therefore,the total number of ways is $30$.

Class 11 Mathematics Permutations and Combinations Circular permutations

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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 cap of one of the three colours red,blue and green,then the number of ways of distributing the caps such that the persons seated in adjacent seats get different coloured caps,is

MHT CET 2023 All MHT CET

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$30$
B
$15$
C
$60$
D
$40$

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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 cap of one of the three colours red,blue and green,then the number of ways of distributing the caps such that the persons seated in adjacent seats get different coloured caps,is

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