Number of pizza topping combinations It seems there are lots of pizza questions but I'm not sure how to apply the answers to my problem.  Obviously I'm not a mathematician.  Essentially I'm trying to determine how many different variations of pizzas there are given the following parameters.  You can choose unlimited, unique toppings (no double toppings).  Each pizza can have at most 1 sauce but the rest can be mixed and matched to your hearts content.  How do I do this?
6 sauces
7 cheeses
15 vegetables
7 meats
2 seasonings    
 A: You can choose the sauce in $\binom{6}{1} + 1$ ways (the $1$ is for no sauce). Then consider filling up $7+15+7+2$ blanks with either $0$ or $1$, $1$ if you want that topping, $0$ if you don't want it. This can be done in $2^{31}$ ways. So the total number of ways is
$$
(\binom{6}{1} + 1) \cdot 2^{31} = 7 \cdot 2\;147\;483\;648 = 15\;032\;385\;536
$$
A: $1$ sauce out of $6 \implies {6 \choose 1} = 6$ ways.
Total remaining toppings are $31$. Each topping can be chosen or not ($2$ ways). This accounts for $2^{31}$ ways.
By multiplication rule, total ways = $6 \cdot 2^{31} = 12884901888$.
A: Well,  for this you break it up into the two types:  Sauces and not sauces.  The answer is slightly different depending on whether no sauce at all is an option (i.e. is it at most 1,  or exactly 1?)
So,  if it's exactly 1,  you have 6 choices for sauce, if it's at most 1, you have 7.   That's independent of the other choices,  so you multiply 6 (or 7) by the rest.
Now,  since the rest of the toppings it's irrelevant how many you have of which variations,  all you need to consider for each individual item (each cheese, veggie, etc), is whether its ON or OFF.   In other words,  each other thing has 2 states,  and each choice is independent of each other, so again the multiplication principle applies.   To simplify the math, you can just add the other things together,  so $7+15+7+2=31$.   Then you have $6*2^{31}$ choices, or $7*2^{31}$ if no sauce is a choice
Basically, divide everything up into completely independent choices, then multiply each outcome of independent events.
A: $7$ * $2^{31}$
Sauce is optional.  Can have doesn't mean must have.  For example, if you said a man can only have 1 wife, that doesn't mean he must marry.  He can stay single.
