Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 60 / 2 / i / a / Solution

Every conditional output distribution is a permutation of $q$. Since Shannon entropy is invariant under a permutation, its value is the same for every input symbol. Thus, for any input distribution,All Shannon entropies in this solution are in bits. A different fixed logarithm base changes every entropy and capacity by the same constant factor.