Five persons A, B, C, D and E are seated in a | vedclass

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(A) Let the colours be $R, B, G$. The number of ways to colour the vertices of a cycle graph $C_n$ with $k$ colours such that no two adjacent vertices

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(A) Let the colours be $R, B, G$. The number of ways to colour the vertices of a cycle graph $C_n$ with $k$ colours such that no two adjacent vertices have the same colour is given by the chromatic polynomial $P(C_n, k) = (k-1)^n + (-1)^n(k-1)$.Here,$n = 5$ (number of persons) and $k = 3$ (number of colours).Substituting the values:$P(C_5, 3) = (3-1)^5 + (-1)^5(3-1)$$= 2^5 - 2$$= 32 - 2$$= 30$Thus,there are $30$ ways to distribute the hats.

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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