Let be a permutation of the integers 0, 1, 2, ... (2n - 1) such that (m) gives the permuted value of m, 0m<2n. Put another way, maps the set of n-bit integers into itself and no two integers map into the same integer. DES is such a permutation for 64-bit integers. We say that has a fixed point at m if (m) = m. That is, if is an encryption mapping, then a fixed point corresponds to a message that encrypts to itself. We are interested in the probability that has no fixed points.

Show the somewhat unexpected result that over 60% of mappings will have at least one fixed point.