Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?

Let $G = (V, E)$ be a random graph with $n = 8$ vertices, where each possible edge exists independently with probability $p = 1/2$.

A cycle of length 3 (a triangle) is uniquely defined by choosing a set of 3 vertices out of the 8 vertices.

1. The total number of triplets of vertices is:

   $$\binom{n}{3} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$$

2. For any specific triplet of vertices $\{u, v, w\}$, a cycle of length 3 exists if and only if all three edges $(u,v)$, $(v,w)$, and $(w,u)$ are present in the graph. Since each edge exists independently with probability $p = 1/2$:

   $$P(\text{cycle on } \{u, v, w\}) = p^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$$

3. By linearity of expectation, the expected number of cycles of length 3 is:

   $$E[X] = \binom{8}{3} \times \left(\frac{1}{2}\right)^3 = 56 \times \frac{1}{8} = 7$$

Hence, the correct option is (C).