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In how many ways can this diagram be coloured subject to the following two conditions? 

(i) Each of the smaller triangle is to be painted with one of three colours: red, blue or green. 

(ii) No two adjacent regions have the same colour.

1 Answer

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These conditions are satisfied exactly when we do as follows: First paint the central triangle in any one of the three colours. Next paint the remaining 3 triangles, with any one of the remaining two colours. 

By the fundamental principle of counting, this can be done in 3 × 2 × 2 × 2 = 24 ways.

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