Expected complete groups of four in a random graph

Twelve bulbs are arranged on a circle and every pair is joined by a wire. Each wire is independently live with probability $\tfrac12$. Call a set of bulbs fully connected if every pair inside it is joined by a live wire. What is the expected number of fully connected sets of exactly $4$ bulbs?

$\tfrac{495}{64}$. There are $\binom{12}{4}=495$ sets of four bulbs, and a given set is fully connected when all of its $\binom42=6$ wires are live, which has probability $2^{-6}$. Linearity of expectation gives $495/64$.