34 million pizzas is a massive understatement

During a baseball game today, heard a commercial for Domino’s Pizza claiming that they have 34 million different pizzas. This is apparently a claim that Domino’s started making in saying that they shouldn’t be forced to list calorie counts: see the Washington Post from June 2012.

In any case, I wondered where they got that number. From their online ordering tool I find:

• on the first page (size & crust): 10 combinations of size and crust;
• on the second page (cheese & sauce): six different amounts of cheese (none, light, normal, extra, double, triple) and thirteen sauce choices: four kinds of sauce each in three different amounts, or none;
• on the third page, 25 different toppings, each of which you can order in six different amounts (none, light, normal, extra, double, triple).

The “34 million” number is presumably 225 = 33,554,432 (yes or no for each topping), but a more accurate calculation of the number of possible pizzas is $10 \times 6 \times 13 \times 6^{25}$ or about $2.2 \times 10^{22}$. Call this number $N$. And even that’s ignoring that you can make independent choices for the two halves of your pizza – so if we really wanted to inflate the number, there are $N + {N \choose 2}$ possible pizzas, where the ${N \choose 2}$ pizzas are just pairs of choices for the two halves. This is around $2.4 \times 10^{44}$.

But nobody would have believed that. And how many of those pizzas are any good?

4 thoughts on “34 million pizzas is a massive understatement”

1. Excellent, but like it is with many real-life food counting problems¹, there’s more to the story.

Having coincidentally ordered from Domino’s last night (there’s a 50%-off promotion through tomorrow for online orders), and having compulsively explored their online ordering tool in the past, I know that you can’t choose any of the 6 topping levels for each of the toppings. There’s a limit of about 10 topping units per pizza, where double counts as 2, triple as 3, etc. (In reality, double and triple aren’t truly two and three times as much topping — at least not at my shop — and I haven’t tested the ordering tool to see how Light and Extra count towards the limit.)

No doubt the actual number of choices will still exceed 34 million — and I think you’re right about where that number came from — but instead of $6^25$ topping possibilities, a closer-to-reality answer is to use the number of solutions to the inequality 0 ≤ a+b+c+d+…+y ≤ 10 where each variable has a value in {0,0.5,1,1.5,2,3}. Which is a fun problem to solve, and I won’t, because I don’t want to take away the pleasure from any readers!

Also worth noting: The number of options varies from location to location. My local shop offers eight crust/size combinations (because their oven isn’t extra-large, I guess), 15 sauce combinations (BBQ inexplicably comes in five different amounts, unlike the others, which come in three) and 26 toppings.

Finally, for anyone who’d prefer a more data-oriented project than the counting problems here, there’s the matter of Domino’s pricing. Someone should make a map or turn it into a GIS or database project. There’s no menu with prices, so you can’t find out what a pizza costs without trying to order it, and the pricing varies considerably across the country. Fortunately it seems to be as cheap as anywhere where I live, and multiple toppings don’t seem to cost much extra, if at all.

2. Rob says:

I’ll have one of each please. If my order is not filled in 30 minutes or if my order arrives cold, is it free?