Mutual Information, is this calculation correct or should I contact the author about fixing it? Background information

According to Cover's text book on information theory  Mutual Information is calculated as: 

$$(1)  I(W;C) = \Sigma_k\Sigma_i P(C_k,W_i)log(\frac{P(C_k,W_i)}{P(C_k)P(W_i)})
$$

Now if we were interested in a single Word W_i we would just calculate this with one sum:

$$(2)   I(W_i;C) = \Sigma_k P(C_k,W_i)log(\frac{P(C_k,W_i)}{P(C_k)P(W_i)})
$$

OR we can rewrite this as 

$$ (3)   I(W_i;C) = \Sigma_k P(C_k,W_i)log(\frac{P(W_i|C_k)}{P(W_i)})
$$

because 
$$ (4)  P(W_i|C_k) = \frac{P(W_i,C_k)}{P(C_k)}$$ (unless i am incorrect)

Question
While reading a paper on feature selection using information theory measurements I came across A segment of the text: 



Should the equation be more like (3), and I should contact the Author, or have I mis-understood/interpreted mutual information?
No there is nothing wrong with the formulas in the paper.  


MI(S) is the average mutual information of set S over all classes c in C. It is NOT a reformulation of the formula for MI(W,C).
MI(w) is identical to your formula for I(Wi,C)  (except it uses slightly different notation).