Pizza

In the above picture we consider the situation where we have four toppings (cheese, pepper, onion, and mushroom) to choose from. There are sixteen possible pizza topping combinations if there are four toppings. If we restrict our attention to just the first two lines in the left column, we can see that if the only topping was cheese, we would have 2 possible pizzas; 1 with only sauce and 1 with cheese. Add the next two rows and we have our plain sauce and cheese options either without pepper, or with pepper for a total of 4 choices. If we add a third option of onion, we have the first 8 combinations by taking the first four and either not adding onion to them or adding onion to them. Again, we can see our pattern of squares emerging from our binary choice of either choosing not to include an ingredient or choosing to include it.

What about each individual number in any given row of Pascal's triangle? What would the 1, 4, 6, 4, and 1 represent for the pizzas with 4 ingredient choices? In our previous Towers solution, we observed that each consecutive number in the triangle corresponded to the number of towers with a certain number of any given color. Here, our choice doesn't represent a color but whether or not a topping is represented on a pizza option. This means we should expect to have 1 pizza with 0 toppings, 4 pizzas that only have 1of each topping, 6 pizzas with 2 of any combination of the toppings, 4 pizzas with 3 of any combination of the toppings, and 1 pizza with all 4 toppings. These are combinations! This reminds us that each number in a row of Pascal's triangle represents n choose r where r is an integer between 1 and n.