# Number of 'Tri - Coloured Triangles' in a random graph.

I am attempting some practice questions where no answers are given. Would it be possible to verify my solution for this particular problem?

Let RBG be a coloured random graph with $$n$$ vertices $$\{1, 2, · · · , n\}$$, where every pair of vertices is connected either by a red edge (with probability $$p$$) or by a blue edge (with probability $$q$$) or by a green edge (with probability $$r$$), such that $$p + q + r = 1$$. (Note that all edges are present, but each edge carries exactly one of three different colours - red, blue or green)

A set of three vertices $$\{i, j, k$$} is said to form a “tri-coloured triangle” if the triangle formed by $$\{i, j, k\}$$ has exactly one red edge, exactly one blue edge and exactly one green edge. Let $$N_3$$ denote the number of tri-coloured triangles in the coloured random graph RBG. Calculate $$E(N_3)$$

My approach is as follows

Given a triangle, there are $$3! = 6$$ ways of colouring the triangle such that each edge has a distinct colour.

Therefore, $$P(\text{A triangle is tri - coloured}) = 6pqr$$.

There are $${n}\choose{3}$$ ways of choosing 3 vertices in the random graph.

Therefore, $$E(N_3) =$$ $${n}\choose{3}$$ $$6pqr = n(n-1)(n-2)pqr$$

• Perfect, except for five missing full stops. – bof Dec 2 '18 at 4:55
• Thank you, @bof ! :) – dzl Dec 3 '18 at 5:27