Discrete math problem ends up with seemingly impossibly large number. EDIT: Posted a tentative answer to correct my mistake based on TonyK's comment.
I saw this question on a forum and tried to solve it, but I came up with a huge number that made me question my logic.
The problem states: "A cafeteria offers $17$ types of pizza and $5$ types of soda. Bob goes there everyday for lunch, always buying two slices of pizza and one soda. However, he never gets exactly the same thing on two consecutive days (that is, each time, the drink or at least one of the slices is different from what he had yesterday). In how many ways can Bob choose his lunch for the next 30 days if he just returned from a vacation (and so hasn’t been at the cafeteria for the past few days)?"
(I'm assuming here that we don't care about the order, i.e picking first a pepperoni slice and then a mushroom slice is the same as the alternate order.)
On the first day, we have $17 \cdot 17 \cdot 5$ possible combinations of $2$ slices + a soda = $1445$ possible lunches (I'm assuming here that both slices can be of the same kind). 
The way I think about each day after the first is to imagine that the cafeteria offers a huge menu of $1445$ possible lunches. On the first day, Bob can choose any of those $1445$ possible lunches. 
However, on each day after that he can only choose from $1444$ of those.
So the total number of combinations of possible lunches over a $30$ day period is $1445 \cdot 1444^{29}$.
The reasoning seems fine to me, but the number is making me doubt myself...
 A: Based on TonyK's comment, I think I can now answer correctly.
Basically, since we don't care about the permutations for the 2 slices of pizza, the actual number of combinations of 2 slices is $\frac{17^2-17}{2} + 17 = 153$, times $5$ sodas $= 765$.
Each day after that, Bob can choose any of those possible lunches besides one, so $764$. 
Therefore the actual number of possible lunch combinations over 30 days, is $765 * 764^{29}$.
EDIT: The reason for $\frac{17^2-17}{2} + 17 = 153$ is that $17$ of those possible $2$ slice combinations, the ones that are the same kind, are going to be unique; but $17 \cdot 16$ of them, where we're picking $2$ different objects from the same set, are going to be counted twice (once in each possible ordering, AB-BA) for each combination, hence why we divide by $2$.
A: Out of 17 types of pizza slices, we need to select two slices.
Total number of selections : 17C2
Out of 5 sodas, we need to select one.
Total number of selections : 5C1
Since we need to select two pizza slices and a soda, we apply the multiplication rule to get 17C2 × 5C1 selections in total. (Which is equal to 680)
Each day after that (as you correctly pointed out), Bob can choose 1 less selection from our total. Therefore, 679.
The initial 680 selections are possible only for the first day, followed by 679 selections for the remaining 29 days. 
This gives us the final answer: 680 × 679^29.
Edit: This answer is based on the assumption that we cannot select the same slice of pizza twice.
