On November 8, 1993, the world's largest carry-out pizza chain first aired, on national television, this very popular commercial spot, titled "Math Whiz". Most viewers probably found this commercial to be cute. Some maybe remembered from their mathematical background that the large number of possibilities had something to do with permutations, factorials, combinations, or some other long forgotten technique. Perhaps only the authors of this article were so intrigued that they determined whether or not the four-year-old "math whiz" was correct in his calculations.

The commercial derives its conclusion using the combinatorics calculation

where each term added together represents the number of combinations of the given number of toppings chosen, up to five, out of eleven available toppings. The result is then squared since the toppings on the two pizzas need not be the same. However, this analysis yields a value that is nearly twice as large as it should be. For example, if the first pizza is a pepperoni pizza, and the second is a sausage pizza, then the analysis would incorrectly treat this as a different possibility than if the first pizza is a sausage pizza, and the second is a pepperoni pizza. To correct this mistake, this total number of possibilities should be divided by two. However, in each case where both pizzas are identical, there is no need to divide by two, since each of these occurrences appears only once. Thus, for

would better describe the desired number of possibilities for two pizzas. The first term divides

where the first term allows for the selection of two different pizzas out of the

The menu pad supplied by the national headquarters, which is used by the local pizza chef, actually has thirteen choices of toppings, excluding extra-cheese, which cannot be used as one of the five selected toppings. The toppings consist of pepperoni (P), mushroom (M), green pepper (GP), onions (O), ham (H), Canadian bacon (CB), bacon (B), ground beef (GB), Italian sausage (IS), black olives (BO), pineapple (PN), hot pepper rings (R), and anchovies (A). The format of the menu pad appears similar to

and the pizza chef simply puts an

Allowing multiple amounts of a topping greatly complicates the calculations. The easiest way of tackling this updated problem is to consider the menu pad. If in addition to double pepperoni, onions and double Italian sausage are also ordered, then omitting the topping symbols and the blank spaces, the menu pad would look like

Similarly,

would represent triple pepperoni, ground beef, and anchovies. In fact, any five toppings, allowing multiple amounts of a topping, can be represented by all distinguishable permutations of twelve

the number of different pizzas with exactly five toppings is

Likewise, the number of pizzas with exactly four toppings (twelve

Continuing in this manner, the number of possibilities for one pizza, allowing multiple amounts of a topping, is

Using

The "math whiz" should now be informed that he can order a different set of two pizzas each day for over 35 million more days than he originally calculated. However, since he is only four-years-old, maybe he will eventually be able to try all the possibilities!