Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem involving combinations. Howard's Hamburger Heaven sells hamburgers with cheese, relish, lettuce, tomato, mustard, or ketchup. How many different hamburgers can be made that use any 3 of the extras?

Step-by-Step Solution

Answer

20 different hamburgers can be made.

Step 1: Understand the Problem

We need to find how many different hamburgers can be made using any 3 extras from a list of 6 possible extras: cheese, relish, lettuce, tomato, mustard, and ketchup.

Step 2: Identify Combinatorial Approach

Since the order in which we select the extras does not matter and each choice consists of choosing 3 extras out of 6, we use combinations for this problem.

Step 3: Apply Combination Formula

The combination formula 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. Here, \( n = 6 \) and \( r = 3 \).

Step 4: Calculate the Combinations

Substitute the values into the formula: \[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \].

Step 5: Verify the Calculation

To ensure our calculation is correct, verify by listing a few combinations manually or revisit each calculation step. For instance, one possible combination is cheese, relish, and lettuce. Calculate manually others to check, and verify that they equal 20.

Key Concepts

Combinatorial Mathematics

Combinatorial mathematics is a fascinating field that deals with counting and arranging objects. It helps us understand how to calculate the number of possible arrangements or selections within a set. When you think about arrangements of items, you might consider different sequences or groups.
This mathematical approach finds applications in various real-life scenarios, such as finding out how many ways you can outfit yourself from a selection of clothes.
It allows us to see the numerous possibilities without having to list each option manually. Combinatorial mathematics is particularly useful in probability, statistics, and even computer science.

Combination Formula

The combination formula is at the heart of determining the number of ways to select items without regard to order. We use it in situations where we want to choose a subset of items from a larger set, and the order doesn't matter.
The formula for combinations is:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

Here, \( n \) represents the total number of items, and \( r \) is the number of items you want to select.
The symbol \(!\) denotes factorial, meaning you multiply a series of descending natural numbers. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
This formula is perfect for scenarios like our hamburger toppings example, where the goal is to identify how many unique sets of extras can be selected from a given list of options.

Permutations and Combinations

Permutations and combinations are key concepts in combinatorial mathematics that help solve problems of arranging and selecting objects.
Although they are related, they are distinct in their applications:

• **Permutations** take the order of items into account. That means each different order of a selection counts as a different outcome. This is ideal when arranging books on a shelf or characters in a password, where sequence matters.
• **Combinations** ignore the order. It's all about choosing groups of items where the sequence isn't important, just like picking our hamburger toppings.
  Whether we pick cheese, tomato, and lettuce or lettuce, tomato, and cheese, it's the same combination.

Understanding when to use combinations or permutations is crucial in solving mathematical problems efficiently and correctly.