Problem 59 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Hamburgers A hamburger chain gives their customers a choice of ten different hamburger toppings. In how many different ways can a customer order a hamburger?

Step-by-Step Solution

Answer

There are 1024 different ways to order a hamburger with the given toppings.

Step 1: Understanding the Problem

To determine the number of different ways a customer can order a hamburger with ten different toppings, we need to consider that the customer can choose any subset of these toppings. This means the customer can choose anywhere from none to all ten toppings.

Step 2: Applying the Counting Principle

Since each topping can either be included or not included, this creates a situation where for each topping there are 2 choices: either to include it or not. This is a classic example of applying the counting principle where each option (inclusion or exclusion) is independent of others.

Step 3: Calculating the Total Number of Combinations

Since there are 10 toppings and each topping can be selected (or not), we have 2 choices per topping. Therefore, the total number of ways to choose from 10 toppings is calculated by raising 2 to the power of 10, which can be written as \(2^{10}\).

Step 4: Performing the Calculation

Now perform the calculation: \(2^{10} = 1024\). This means there are 1024 different ways the customer can order their hamburger with the different possible combinations of toppings.

Key Concepts

Counting Principles

When faced with problems like determining the number of ways to order a hamburger with a variety of toppings, the concept of counting principles becomes very useful. Counting principles are mathematical rules that help us calculate the total number of possible outcomes without having to list every single possibility. In this case, each topping represents a choice: either to include it on the hamburger or not.

What makes counting principles handy is their ability to manage numerous options efficiently. By understanding that for each topping there are two possible choices, and every choice is independent from the others, we employ the basic counting principle. This principle is the foundation for understanding more complex combinatorial problems.

• Independence of choices: Each decision does not affect the others.
• Binary choice per item: You can choose to either include or exclude a topping.

Mastering counting principles is essential in combinatorics because it lays the groundwork for understanding more advanced topics such as permutations and combinations.

Combinations

Combinations are a key concept in combinatorics that differ from permutations by focusing on selecting items without considering order. This makes combinations ideal for problems where the arrangement of selected items does not matter, like in our hamburger toppings situation.

When you have a set of items, such as toppings in this case, and you wish to select a subset of them in any manner, combinations help you calculate how many such subsets are possible. In the scenario of picking toppings, you can choose any number of these from none to all, hence 0 to 10.

• Subsets: Combinations deal with choosing any number of items from a set, ignoring the sequence.
• Formula: The number of ways to choose \(r\) items from \(n\) is \(\binom{n}{r}\), which reads as "n choose r.”

In our hamburger example, although we did not directly use "n choose r", understanding the nature of combinations provides insight into how this variety of choices adds up to numerous possibilities.

Binomial Theorem

While the binomial theorem isn't directly applied in the hamburger toppings problem, it provides the mathematical foundation for understanding how expressions expand when each component has two choices—much like each topping can be chosen or not.

The binomial theorem expresses the expansion of a binomial raised to any power, which ties back to the expression \((x+y)^n\). Here, each term in the expansion corresponds to a way to pick 0 to \(n\) items (where \(n\) is the total possible items, like the toppings). Each combination is represented by the coefficients \(\binom{n}{k}\), known as binomial coefficients.

• Application: Represents combinations with two options (like with toppings, for including or not including a given topping).
• Relevance: Provides the theoretical underpinning for why \(2^{10} = 1024\) nicely demonstrates the summation of all possible combinations of ten toppings.

Understanding the binomial theorem helps students see the bigger picture in combinatorics, illustrating how everything fits into a cohesive mathematical universe.