Applying the Product Rule, but getting two diffent answers 
Three officers—a president, a treasurer, and a secretary—are to be chosen
  from among four people: Ann, Bob, Cyd, and Dan. Suppose that, for various
  reasons, Ann cannot be president and either Cyd or Dan must be secretary.
  How many ways can the officers be chosen?

There are three choices for president (all except Ann), three choices for treasurer (all except the one chosen as president), and two choices for secretary(Cyd or Dan). Therefore, by the multiplication rule, there are $3 \cdot 3 \cdot 2 = 18$ choices in all.
But if we choose  the secretary first, then the president, then the treasurer, the answer is $8$.
Why do we get two different answers here?
 A: Answer corrected following the helpful commentary by 6005
Considering all possibilities there are eight cases:
$$\begin{array}{|c|c|c|} 
\hline \text{T} & \text{S} & \text{P} \\ \hline
\text{A} & \text{C} & \text{B} \\ \hline
\text{A} & \text{C} & \text{D} \\ \hline
\text{A} & \text{D} & \text{B} \\ \hline
\text{A} & \text{D} & \text{C} \\ \hline
\text{B} & \text{C} & \text{D} \\ \hline
\text{B} & \text{D} & \text{C} \\ \hline
\text{C} & \text{D} & \text{B} \\ \hline
\text{D} & \text{C} & \text{B} \\ \hline
\end{array}$$
(T stands for treasurer, S for secretary, P for president, A for Ann, B for Bob, C for Cyd and D for Dan.)
A: 
There are $3$ choices for president (all except Ann), $3$ choices for treasurer (all except the one chosen as president), and $2$ choices for secretary (Cyd or Dan). Therefore, by the multiplication rule, there are $3 \cdot 3 \cdot 2=183 \cdot 3  \cdot 2=18$ choices in all.

This reasoning is not correct, because there aren't necessarily $2$ choices for secretary. There could be no choices (if Cyd and Dan were chosen as president and treasurer), or $1$ choice (if one of them was chosen as president or treasurer), or $2$ choices (if neither of them was chosen as president or treasurer).
You have to be careful when applying the multiplication (product) rule. The rule says that the number of choices for $A$ and $B$ is equal to the number of choices for $A$ times the number of choices for $B$ given the choice for $A$ -- BUT it only applies if the number of choices for $B$ does not depend on which choice for $A$ was made.
The correct answer is $8$. Like you suggested, we can get this by choosing secretary first, then president, then treasurer. There will be $2$ choices for each, and fortunately this number will not depend on previous choices, so the multiplication rule applies.
