Note that a pizza with cheese and tomatoes is the same thing as a pizza with tomatoes and cheese.
1585
Practice makes perfect
We have a group of 12 toppings and want to know how many pizzas we can create when choosing 1, 2, 3, 4, and 5 toppings. Note that a pizza with cheese and tomatoes is the same thing as a pizza with tomatoes and cheese.
Pizza & Cheese ⇔ Cheese & Pizza
Therefore, we are looking for the number of combinations, _nC_r, we can create with five or fewer toppings.
12 toppings, choose 1: _(12)C_1 [0.4em]
12 toppings, choose 2: _(12)C_2 [0.4em]
12 toppings, choose 3: _(12)C_3 [0.4em]
12 toppings, choose 4: _(12)C_4 [0.4em]
12 toppings, choose 5: _(12)C_5
Let's calculate the first combination.
If we add one more topping to the cheese, we can create 12 more pizza types. Let's repeat the calculations with _(12)C_2, _(12)C_3, _(12)C_4, and _(12)C_5 to determine how many combinations we can form when adding 2, 3, 4, and 5 toppings to the pizza.
_(12)C_2:& 12!/2!(12-2)!= 66
_(12)C_3:& 12!/3!(12-3)!=220
_(12)C_4:& 12!/4!(12-4)!=495
_(12)C_5:& 12!/5!(12-5)!=792
By adding all of the combinations we get the how many different pizzas we can create with five or fewer toppings.
12+66+220+495+792=1585
