Combinations question

[collegekid]

A pizza restaurant lists the following options on its menu:
Crust: Thick, hand-tossed, or thin
Meat: Sausage, pepperoni, hamburger, or Canadian bacon
Veggies: Olives, onions, or mushrooms
Customers must choose one type of crust for each pizza, and may choose to add one or more of the toppings.
a.How many different pizzas can be ordered by choosing one item from each category?
b.How many different vegetarian pizzas can be ordered (for this problem, we’ll consider a “plain” cheese pizza to be vegetarian)?
c.How many different pizzas with up to three toppings can be ordered?
d.How many different pizzas can be ordered at this restaurant?

a. i got 21
b. 35
c. 35
d. 128
can anyone verify if this is right?

 

[soroban]

Hello, collegekid!

Your answers are way off . . . Sorry!

A pizza restaurant lists the following options on its menu:
. . Crust: Thick, hand-tossed, or thin
. . Meat: Sausage, pepperoni, hamburger, or Canadian bacon
. . Veggies: Olives, onions, or mushrooms

Customers must choose one type of crust for each pizza, and may choose to add one or more of the toppings.

a. How many different pizzas can be ordered by choosing one item from each category?

3 choices of crust, 4 choices of meat, 3 choices of veggies.

There are: .\(\displaystyle 3\times 4 \times 3 \:=\:36\) pizzas of this type.

b.How many different vegetarian pizzas can be ordered?
For this problem, we’ll consider a “plain” cheese pizza to be vegetarian.

For each of the 3 toppings, there are 2 choices: include it or omit it.
. . Hence, thyere are: .\(\displaystyle 2^3\:=\:8\) possible choices for vegetables.

With 3 choices of crust, there are: .\(\displaystyle 8 \times 3 \:=\:24\) vegetarian pizzas.

c.How many different pizzas with up to three toppings can be ordered?

There are 7 toppings available.

\(\displaystyle \text{There are: }\;\begin{array}{ccccc} _7C_0 &=& 1 & \text{pizza with 0 toppings }\\ _7C_1 &=& 7 & \text{pizzas with 1 topping } \\ _7C_2 &=& 21 & \text{pizzas with 2 toppings} \\_7C_3 &=& 35 & \text{pizzas with 3 toppings} \end{array}\)

So there are: .\(\displaystyle 1 + 7 + 21 + 35 \:=\:64\) such pizzas.

With 3 choices of crusts, there are: .\(\displaystyle 3 \times 64 \:=\:192\) possible pizzas of this type.

d.How many different pizzas can be ordered at this restaurant?

There are 7 available toppings.
For each there are 2 choices: include it or omit it.
. . There are: .\(\displaystyle 2^7 \:=\:128\) choices of toppings.

With 3 choices of crust, there are: .\(\displaystyle 3 \times 128 \:=\:384\) possible kinds of pizza.