Domino's Pizza Claims 75 Million Pizza Types

Arne Olav Hallingstad
27th February 2011
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Introduction

I saw an ad in the Evening Standard (newspaper) on the train on the way home from work the other day saying:

Someone once calculated that there are 75 million ways to customise a Domino's pizza. So what'll it be tonight then?
...

Is this really true? 75 million combinations? Let's go to their home page and see. We'll be looking at the Dominos UK store for the London area.

Finding the Options

Dominos has a note on their site saying the exact selection of toppings, sauces, pizza bases and special pizza offers may be different from store to store, but overall I would guess it's mostly the same selection given that it's franchise.

We will ignore anything but the pizza section on their web site. That means we won't count any of these sections: offers, pizza sides (chicken wings, garlic mozzarella sticks, et cetera), desserts, drinks, subs, catering.

Here's a picture of the pizza section:

We'll only be looking at the "create your own" pizza as all the pizza types can be created from the customisation options.

Unique Combinations

We see that we have 5 sections to choose a certain number of combinations in.

Pizza Size

Must choose one out of 4 pizza sizes.

Pizza Base

Must choose one out of 4 pizza bases.

Sauces

For sauce, cheese and topping there's a choice of either "single topping" or "double topping" for each item.

single topping: 3 + no choice at all giving 4 choices.
double topping: 3 * 2 + no choice at all giving 7 choices.

Cheese

Single topping: For the choice of cheese we have 2 + no cheese at all giving 3 choices.
Double topping: For the choice of cheese we have 2 * 2 + no cheese at all giving 5 choices.

Additional Toppings

There's a total of 22 toppings to choose from, but depending on us counting double topping as a valid choice as well it becomes 44 choices.

We can choose up to 9 toppings on a pizza. We'll have to add up all the possible combinations of toppings given that we select 0, 1, and so on up to 9 toppings out of the 22 or 44 choices.

Single Topping

We use the Binomial coefficient to find all the possible combinations of the 22 choices toppings:

combination( 22, 0 ) = 1
combination( 22, 1 ) = 22
combination( 22, 2 ) = 231
combination( 22, 3 ) = 1540
combination( 22, 4 ) = 7315
combination( 22, 5 ) = 26334
combination( 22, 6 ) = 74613
combination( 22, 7 ) = 170544
combination( 22, 8 ) = 319770
combination( 22, 9 ) = 497420
Total: 1 097 790

Double Topping

Let's do the calculation again where we count the double topping, giving a total of 44 choices.

combination( 44, 0 ) = 1
combination( 44, 1 ) = 44
combination( 44, 2 ) = 946
combination( 44, 3 ) = 13244
combination( 44, 4 ) = 135751
combination( 44, 5 ) = 1086008
combination( 44, 6 ) = 7059052
combination( 44, 7 ) = 38320568
combination( 44, 8 ) = 177232627
combination( 44, 9 ) = 708930508
Total: 932 778 749

Close to a billion combinations just for the toppings, that's already way above the 75 million in the ad.

Conclusion

Total pizza combinations:

Single topping only: 4 * 4 * 4 * 3 * 1 097 790 = 210 775 680 pizza combinations
Single/Double toppings: 4 * 4 * 7 * 5 * 932 778 749 = 522 356 099 440 pizza combinations

Given that the single topping and double topping basically gives us the same pizza, except for the amount given, I'd consider only the single topping as a choice that counts. That means the actual number of pizza combinations at Dominos Pizza is close to 211 million, and not the 75 million the ad is telling us!

Perhaps the people making the ad didn't count the pizzas you could choose without any sauce or cheese, which I admit would be a pretty boring pizza to eat, which would make the actual number of pizza combinations:

Single topping only: 4 * 4 * 3 * 2 * 1 097 790 = 105 387 840 pizza combinations
Single/Double toppings: 4 * 4 * 6 * 4 * 932 778 749 = 358 187 039 616 pizza combinations

There's about 105 million types of pizzas to choose from at Dominos, not 75 million which the ad says. As an organization worth 750 million pounds I'm disappointed in Dominos not verifying this 75 million figure calculated by "someone".