Problem 48 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Three-Topping Pizzas A pizza parlor offers a choice of 16 different toppings. How many three-topping pizzas are possible?

Step-by-Step Solution

Answer

There are 560 different three-topping pizzas possible.

Step 1: Understand the Problem

We need to calculate how many different combinations of three toppings can be made from a possible 16 toppings.

Step 2: Define Combinations

Combinations refer to selecting items from a larger pool where order does not matter. The mathematical formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Substitute Values

In this problem, \( n = 16 \) and \( r = 3 \). Substitute these values into the combinations formula: \( C(16, 3) = \frac{16!}{3!(16-3)!} \).

Step 4: Simplify the Formula

Calculate \( 16! \) divided by \( 3! \times 13! \). This simplifies to \( \frac{16 \times 15 \times 14}{3 \times 2 \times 1} \).

Step 5: Perform the Calculation

Now, calculate the values: \(16 \times 15 \times 14 = 3360 \) and \( 3 \times 2 \times 1 = 6 \). Divide them: \( \frac{3360}{6} = 560 \).

Step 6: Interpret the Result

There are 560 different ways to choose three toppings from 16 available toppings.

Key Concepts

Permutations

Permutations deal with the arrangement of a set of objects where the order matters. In different situations, you might want to know how many ways you can organize a group of items. For example, if you have three different toppings for pizza and you must decide the order of placing them, permutations will be helpful. To calculate permutations, you use the formula: \[ P(n, r) = \frac{n!}{(n - r)!} \]

• \( n \) represents the total number of objects available.
• \( r \) is the number of objects you want to arrange.

Let's say if you were asked to arrange three toppings out of 16 in a row, you would use permutations. But in our pizza example, we don't care about the order, so we'll focus on combinations instead.

Factorials

Factorials are crucial for understanding concepts like permutations and combinations because they help calculate the total possible arrangements of a set. The factorial of a number \( n \), written as \( n! \), means multiplying the number \( n \) by every positive integer less than itself.For instance, if you want to calculate \(4!\): \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]Factorials grow very fast, making them useful in combinatorics problems. In our pizza example, calculating the number of three-topping combinations involves using \( n! \), \( r! \), and \((n-r)!\). Understanding how factorials work helps break down these seemingly complex operations into manageable steps.

Combinatorics

Combinatorics is the mathematical study of counting, arranging, and structuring objects. It is widely used in problems where the goal is to determine the number of ways to select, arrange or order items. In the context of our exercise, combinatorics helps determine how many different combinations of toppings can be chosen for a pizza when order doesn't matter.

• **Combinations** allow calculating the number of ways to pick items from a larger set without regard to the order.
• The formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \), providing the needed count for our pizza toppings problem.

In our example, using the formula with 16 toppings, calculating \( C(16, 3) \), shows there are 560 unique ways to choose three toppings, utilizing the combinatorics principles.