Question

A local pizza parlor has the following list of toppings available for selection. The parlor is running a special to encourage patrons to try new combinations of toppings. They list all possible three topping pizzas (3 distinct toppings) on individual cards and give away a free pizza every hour to a lucky winner. Find the probability that the first winner randomly selects the card for the pizza topped with spicy italian sausage, banana peppers and beef. Express your answer as a fractionPizza toppings: Green peppers, onions, kalamata olives, sausage, mushrooms, black olives, pepperoni, spicy italian sausage, roma tomatoes, green olives, ham, grilled chicken, jalapeño peppers, banana peppers, beef, chicken fingers, red peppers

Answer 1

The probability that the first winner randomly selects the card for the pizza topped with spicy italian sausage, banana peppers, and beef is \( \frac{1}{680} \).

Step-by-step explanation:

To calculate the probability that the first winner randomly selects the card for the pizza topped with spicy italian sausage, banana peppers, and beef, we first need to determine the total number of possible distinct three-topping pizzas that can be made from the list provided.

• The toppings available are: Green peppers, onions, kalamata olives, sausage, mushrooms, black olives, pepperoni, spicy italian sausage, roma tomatoes, green olives, ham, grilled chicken, jalapeño peppers, banana peppers, beef, chicken fingers, and red peppers.

There are a total of 17 toppings to choose from. Since we want to choose 3 distinct toppings for a pizza, we use the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \) where \( n \) is the total number of items to pick from, and \( k \) is the number of items to pick. Here, \( n=17 \) and \( k=3 \).

\( C(17, 3) = \frac{17!}{3!(17-3)!} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = 17 \times 8 \times 5 = 680 \).

So, there are 680 different distinct three-topping pizza combinations that can be made. Since only one of these combinations includes spicy italian sausage, banana peppers, and beef, the probability that this specific combination is chosen is \( \frac{1}{680} \).

Therefore, the probability that the first winner selects the card for the pizza with spicy italian sausage, banana peppers, and beef is \( \frac{1}{680} \).

Answer 2

First, we need to find out how many possible combinations of pizza toppings there would be.

To do this, we will use the formula for Combination.

Combination is all the possible arrangements of things in which order does not matter. In our example, this would mean that a pizza topped with spicy Italian sausage, banana pepper, and beef is the same as a pizza topped with banana pepper, beef, and Italian sausage.

The formula for combination is

`C(n,r)=^nC_r=_nC_r=(n!)/(r!(n-r)!)`

From our given, n would be 17, since there are a total of 17 toppings (including spicy Italian sausage, banana peppers, and beef) and r would be 3 since there are three toppings that you chose.

Substituting it in the formula,

`C(n,r)=(n!)/(r!(n-r)!)`
`C(17,3)=(17!)/(3!(17-3)!)`
`C(17,3)=680`

Now, since we know that there are a total of 680 combinations of pizza toppings, we can now solve the probability of the first winner selecting a pizza topped with Italian sausage, banana peppers, and beef.