Solution
= Solution
For $S_i=[n]\setminus\{i\}$, the <chain rule for information entropy> gives
$$
H(X_{S_i})=\sum_{j\in S_i}H(X_j\mid X_{S_i\cap[j-1]}).
$$
Each 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:
$$
\sum_{i=1}^nH(X_{S_i})
\geq(n-1)\sum_{j=1}^nH(X_j\mid X_1^{j-1})
=(n-1)H(X_1^n).
$$
This is the required special case of <Shearer's inequality>.