# Expected number of monocolor triangles in random 2-coloring

So I have to color the edges of $K_6$ either red or blue. Let $X$ be the random variable that assigns to a coloring the number of monocolor triangles that can be found. (By monocolor triangle I mean that all the edges of a given triangle are either all blue or all red). Then this is what I did. First I counted the number of triangles in $K_6$, there are $\binom{6}{3}=20$ of this. Let $U_1,...,U_{20}$ be all the distinct triangles in $K_6$ Define $X_i$ to be the number of monocolors triangles in $U_i$, so it is $1$ if $U_i$ has the edges all of the same color, and zero otherwise.

Hence, $E[X]=E[X_1+...+X_{20}]$ which by linearity of expectation we have that is is equal to $E[X_1]+...+E[X_{20}]$. For a given $U_i$, we have that $$E[X_i]=1\cdot P(U_i\text{ is monocolor})=2\cdot\frac{1}{2^3}=\frac{1}{4}$$Hence, we have that the solutions is $20\cdot \frac{1}{4}=5$.

Is the above correct? It seems like I have the wrong answer according to the answer key.

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It looks correct to me. What does the answer key say? – Robert Israel Jun 12 '12 at 20:23
$15/4$${}{}{}{}{}{}$ – Daniel Montealegre Jun 12 '12 at 20:28
I just enumerated all possibilities on computer and summed up: the answer is 5 as both you and Robert say. – Jack Schmidt Jun 12 '12 at 20:31
Well I guess it is a typo then. Thanks to both of you. – Daniel Montealegre Jun 12 '12 at 21:01