Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 224 / 3 / c

Draw $X_1^n$ uniformly from $B$ and let $J$ be uniform on $\{1,\dots ,n\}$ independently. ThenIf $P_i$ denotes the marginal law of $X_i$, this also says $P_B=n^{-1}{\sum }_i{P}_i$. Since $X_1^n$ is uniform on $B$,Subadditivity of information entropy followed by concavity of information entropy givesExponentiating proves $|B|\le 2^{nH(P_B)}$.