Suppose Kn is a complete graph whose vertices are indexed by [n] = {1, 2, 3, ..., n}, where n 4. In this question, a cycle is identified solely by the collection of edges it contains; there is no particular orientation or starting point associated with a cycle. (Give your answers in terms of n for the following questions.) (a) How many Hamiltonian cycles are there in Kn? (b) How many Hamiltonian cycles in Kn contain the edge {1, 2}? (c) How many Hamiltonian cycles in Kn contain both the edges {1, 2} and {2, 3}? (d) How many Hamiltonian cycles in Kn contain both the edges {1, 2} and {3, 4}? (e) Suppose that M is a set of k n 2 edges in Kn with the property that no two edges in M share a vertex. How many Hamiltonian cycles in Kn contain all the edges in M? Give your answer in terms of n and k. (f) How many Hamiltonian cycles in Kn do not contain any edge from {1, 2}, {2, 3} and {3, 4}