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    

$1$ 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 = $6 \cdot 2^{31} = 12884901888$.
