A pizza parlor problem How many different large pizzas with

A pizza parlor problem. How many different large pizzas with no double toppings can be made? (for example, mushroom, pepperoni, and sausage is one possibility, and so is mushroom and sausage, but not double mushroom and sausage. Your account should also include the case of no toppings.) you can’t answer the question yet because you don’t know how many toppings the pizza parlor offers. a. Determine the number of different pizzas when there are exactly 3 topping from (g.g.., mushroom, pepperoni, and sausage.). b. Determine the number of different pizzas when there are exactly 4 toppings to choose from. c. Determine the number of different pizzas when there are exactly 5 toppings to choose from. d. Look for a pattern in your answers in parts (a), (b), and (c). Based on the pattern you see, predict the number of different pizzas when there are 10 toppings to choose from. e. Now find a different way to determine the number of different pizzas when there are 10 toppings to choose from. This time, think about the situation in the following way: pepperoni can be either on or off, mushrooms can be either on or off, sausage can be either on or off, and so on, for all 10 toppings, explain clearly how to use this idea to answer the question and why this method is valid.

Solution

a. In case of 3 toppings to choose from,

no. of pizzas = pizza with 0 , 1 , 2 and 3 toppings = 3C0 + 3C1 + 3C2 + 3C3 = 8

b. In case of 4 toppings to choose from,

no. of pizzas = pizza with 0 , 1 , 2, 3 and 4 toppings = 4C0 + 4C1 + 4C2 + 4C3 + 4C4 = 16

c. Similarly, in case of 5 toppings, no. of pizzas = 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 32

General pattern, in case of n toppings, no.of pizzas = 2ⁿ

d. When n=10,

no.of pizzas = 2¹⁰

= 1024

This method is valid because each topping can either on or not on the pizza, so 2 ways for each topping,so 2*2 ways for 2 toppings and so on.