Problem 42 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Pizza Toppings In how many ways can three pizza toppings be chosen from 12 available toppings?

Step-by-Step Solution

Answer

There are 220 ways to choose three pizza toppings from twelve.

Step 1: Identify the Problem Type

The problem involves choosing three toppings from a set of 12 available toppings, which is a combination problem. In combination problems, the order does not matter.

Step 2: Know the Combination Formula

Recall the formula for combinations, which 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 the Given Values

In this exercise, \( n = 12 \) and \( r = 3 \). Substitute these values into the combination formula: \[C(12, 3) = \frac{12!}{3!(12-3)!}\]

Step 4: Simplify the Factorials

Simplify the expression by calculating the factorials:\[12! = 12 \times 11 \times 10 \times 9! \quad\text{and}\quad 9! = \text{cancels out with the denominator},\]\[C(12, 3) = \frac{12 \times 11 \times 10}{3!}\]Calculate \(3! = 3 \times 2 \times 1 = 6\).

Step 5: Complete the Calculation

Now, perform the division:\[\frac{12 \times 11 \times 10}{6} = \frac{1320}{6} = 220\]So, there are 220 ways to choose three toppings from 12 available toppings.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics that often appear in calculations involving permutations and combinations. A factorial is the product of all positive integers up to a given number.
For example, the factorial of 5, denoted as 5!, is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are used in combinations to determine the number of ways items can be selected. They help simplify complex multiplicative expressions.
Factorials grow rapidly with larger numbers, which can lead to very large values even for relatively small integers. Understanding how to work with factorials efficiently, such as through simplification or canceling terms, is key to solving combination problems efficiently.

Combination Formula

The combination formula is an essential mathematical tool for finding the number of ways to choose a subset of items from a larger set, where the order of selection does not matter.
The formula is expressed as:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

Where:

• \( n \) is the total number of items available.
• \( r \) is the number of items to be chosen.

The combination formula uses factorials to account for the total permutations of the items and then adjusts by dividing out the permutations that represent different orders of the same selections.
This makes it particularly useful in situations like choosing pizza toppings, where the order of topping selection doesn't change the outcome.

Problem Solving Steps

Approaching a combination problem methodically improves your chances of solving it correctly.
Here is a clear step-by-step approach you can use:

• Identify the Problem Type: Recognize that the problem involves combinations, meaning the order doesn't matter.
• Recall the Combination Formula: Make sure you are familiar with the formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
• Substitute the Given Values: Input your specific problem values into the formula. For example, if you have 12 items and you need to choose 3, set \( n = 12 \) and \( r = 3 \).
• Simplify the Factorials: Break down large factorials to manageable pieces and cancel out common terms to simplify the calculation.
• Complete the Calculation: Perform the final division to find the number of combinations. For instance, \( \frac{12 \times 11 \times 10}{6} = 220 \) tells us there are 220 ways to choose the 3 items from 12.

By following these structured steps, you harness the power of the combination formula efficiently to solve problems with confidence.