Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 224 / 2 / b / Solution

For $S_i=[n]\setminus \{i\}$, the chain rule for information entropy givesEach summand is at least $H(X_j\mid X_1^{j-1})$ because conditioning reduces entropy. Summing over $i$, each $j$ occurs $n-1$ times:This is the required special case of Shearer's inequality.