= Solution
A <discrete memoryless channel> has finite input and output <alphabets> and fixed <conditional probabilities> $M_{ij}=\mathbb P(Y=j\mid X=i)$, whose rows <sum> to one. Memorylessness means that for successive uses, conditional on the input sequence, the output <probabilities> factor as the product of these one-use <conditional probabilities>.
Its information <Channel capacity>[capacity] in <bits> per use is
$$
\boxed{C=\max_{p_X}I(X;Y)=\max_{p_X}\left[H(Y)-\sum_i p_X(i)H(M_{i\cdot})\right].}
$$
The maximization varies the input distribution while keeping the <Communication channel>[channel] <matrix> fixed. The output distribution is $p_XM$ in the row-vector convention.
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