Pizza Topping combinations I run a pizza joint in Seattle, USA, and would love to know how many different combinations of pie we can create.
We have:


*

*23 toppings

*12 "house" pizzas

*2 sizes (medium and large)

*two different bases (tomato sauce and olive oil)

*the ability to do half and half pizzas


Constraints and rules:


*

*maximum of four toppings per pizza or per side

*house pizzas count as distinct choices

*no substitutions or exclusions on house pizzas

 A: Hopefully your aim is to learn something from this exercise, rather than to simply stick the final number on a flyer. I am providing the following answer under this assumption of good will.
Let us first calculate the number of possible whole (i.e. not half-half) pizzas that you can do. Assuming that a pizza with zero toppings (i.e. a margherita) is valid, we have:
$$N = 2\cdot2\cdot\sum_{n_t=0}^{4} \binom{23}{n_t},$$
where $n_t$ denotes the number of toppings, and the two $2$'s account for the two possible sizes and the two possible bases.
Now for the number of half-half pizzas. We can consider a half-half pizza as a combination of two, non-identical pizzas from the above $N$. However, there is an important detail that we need to take care of: we cannot combine a medium pizza with a large pizza. Therefore, we split $N$ in half to get the number of possible medium or large pizzas. Then, $\frac{1}{2}\frac{N}{2}(\frac{N}{2}-1)$ gives us the number of possible half-half medium or large pizzas. The number of possible medium and large half-half pizzas is thus $\frac{N}{2}(\frac{N}{2}-1)$.
Now the total number of possible pizzas is given by: $$T=12+N+\frac{N}{2}(\frac{N}{2}-1).$$
$T$ turns out to be 475,523,454, i.e. you can do a total of almost half a billion different pizzas.
Assuming that your ovens can cook, say, up to 10 pizzas at a time, and that a pizza takes around 10 minutes to cook, it would take you around 904 years non-stop to make every possible pizza. Good luck with that :-).
A: I'm pretty sure this would just be a product of factorials.
(23!) * (12!) * (2!) * (2!) * (2!) = 9.9065259 * 10^31
That's assuming you don't double up on anything.
EDIT: After reading MGA's answer, I'm pretty sure he's right. I assumed that the 12 would be part of the combination which it shouldn't be. Also, I didn't account for the half pizza reducing the possible combinations of toppings.
