Overview:
To find the expected number of unordered cycles of length three in an undirected random graph of eight vertices, we can use the linearity of expectation. We will count the number of cycles of length three for each vertex and then take the average of these counts.

Linearity of Expectation:
The linearity of expectation states that the expected value of a sum of random variables is equal to the sum of their expected values. In our case, we can consider each vertex as a random variable and count the number of cycles of length three it participates in.

Counting Cycles of Length Three:
To count the number of cycles of length three for a vertex, we need to consider all possible pairs of adjacent vertices and check if there is an edge between them. If there is, then we have a cycle of length three.

Calculating Expected Value:
1. For each vertex, calculate the number of cycles of length three it participates in.
2. Sum up the counts for all vertices.
3. Divide the sum by the total number of vertices (eight) to get the expected number of cycles of length three.

Detailed Calculation:
1. For each vertex, count the number of cycles of length three it participates in:
- The number of cycles of length three for a vertex depends on the number of edges between its adjacent vertices.
- In an undirected random graph, the probability of an edge between a pair of vertices is 1/2.
- For each adjacent pair of vertices, the probability of having an edge between them is 1/2.
- The number of possible adjacent pairs for a vertex is the degree of that vertex.
- The degree of each vertex in an undirected random graph is a random variable following a binomial distribution with parameters (n-1, 1/2), where n is the total number of vertices.
- The expected value of a binomial distribution with parameters (n, p) is n * p. Therefore, the expected degree of each vertex is (n-1) * (1/2) = 7/2.
- We can use the expected degree to calculate the expected number of adjacent pairs for each vertex, which is 7/2 * 2 = 7.
- The probability of an adjacent pair having an edge between them is 1/2. Therefore, the expected number of cycles of length three for a vertex is 7 * (1/2) = 7/2.

2. Sum up the counts for all vertices:
- Since there are eight vertices, the total number of cycles of length three is 8 * (7/2) = 28.

3. Divide the sum by the total number of vertices to get the expected number of cycles of length three:
- The expected number of cycles of length three is 28 / 8 = 7/2 = 3.5.

4. Since the number of cycles of length three must be an integer, we round the expected value to the nearest integer, which is 4.

Therefore, the expected number of unordered cycles of length three in the given undirected random graph is 4, which matches option (c).