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A pizza shop has available toppings of peppers, sausage, pepperoni, bacon, and mushrooms. How many different ways can a pizza be made with 2 toppings?
Answer:

A pizza shop has available toppings of peppers, sausage, pepperoni, bacon, and mushrooms. How many different ways can a pizza be made with $2$ toppings? $\newline$ Answer:

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Experimental probability

Full solution

Q. A pizza shop has available toppings of peppers, sausage, pepperoni, bacon, and mushrooms. How many different ways can a pizza be made with

2

toppings?

\newline

Answer:

Given Toppings Selection: We are given $5$ different toppings to choose from and we want to know the number of different pizzas we can make with exactly $2$ toppings. This is a combination problem because the order in which we select the toppings does not matter. The formula for combinations is $C(n, k) = \frac{n!}{k!(n - k)!}$ , where $n$ is the total number of items to choose from, $k$ is the number of items to choose, $n!$ is the factorial of $n$ , and $k!$ is the factorial of $k$ .
Calculate Total Toppings Factorial: First, we calculate the factorial of the total number of toppings $n$ , which is $5$ . The factorial of a number is the product of all positive integers up to that number. So, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ .
Calculate Toppings Wanted Factorial: Next, we calculate the factorial of the number of toppings we want on the pizza $k$ , which is $2$ . So, $2! = 2 \times 1 = 2$ .
Calculate Remaining Toppings Factorial: We also need to calculate the factorial of the difference between the total number of toppings and the number of toppings we want on the pizza $n - k$ . Since $n$ is $5$ and $k$ is $2$ , $n - k = 5 - 2 = 3$ . So, $3! = 3 \times 2 \times 1 = 6$ .
Use Combination Formula: Now we can use the combination formula to find the number of different pizzas that can be made with $2$ toppings. We plug in our values into the formula $C(5, 2) = \frac{5!}{(2!(5 - 2)!)}$ .
Substitute Factorial Values: Substitute the factorial values we calculated into the formula: $C(5, 2) = \frac{120}{2 \times 6}$ .
Perform Calculations: Perform the calculations: $C(5, 2) = \frac{120}{2 \times 6} = \frac{120}{12} = 10$ .

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