An electrician sets up 12 light bulbs on a circle. Each pair of light bulbs has a wire connecting them. Each individual wire is active with probability 21, independent of all other wires. A collection of 2≤k≤12 light bulbs is said to form a complete k−circuit if between any 2 light bulbs in the circuit, we have an active wire. Find the expected number of 4−circuits in our electrician's arrangement of light bulbs.
