Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

A fast food restaurant advertises that it offers any combination of 8 condiments on a hamburger, thus giving a customer 256 choices. How was this number obtained?

Step-by-Step Solution

Answer

The number of combinations (256) is obtained by \(2^8 = 256\).

Step 1: Understanding the choices for each condiment

Each of the 8 condiments can be either included or not included on a hamburger. This gives each condiment 2 possible choices: "yes" (include it) or "no" (do not include it).

Step 2: Calculating total combinations

To find the total number of combinations, we calculate the product of the individual choices for each condiment. Since each of the 8 condiments has 2 choices, the total number of combinations is given by the expression \(2^8\).

Step 3: Expanding the power

Calculating \(2^8\) involves multiplying 2 by itself 8 times. So, \(2^8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\).

Step 4: Solving the expression

Carrying out the multiplication: \(2 \times 2 = 4\), \(4 \times 2 = 8\), \(8 \times 2 = 16\), \(16 \times 2 = 32\), \(32 \times 2 = 64\), \(64 \times 2 = 128\), and finally \(128 \times 2 = 256\). Thus, \(2^8 = 256\).

Key Concepts

Binary Choice

In the context of combinatorics, a binary choice refers to the decision-making process involving two distinct options. For each decision point, you choose either one option or the other.
In our hamburger problem, each condiment acts as a binary choice — you can either include it or skip it. This concept is easy to grasp because it’s much like a simple yes-or-no question.
Binary choices are fundamental in many areas, including computer science and decision-making processes. When you're faced with a binary choice, you multiply the number of choices by 2 to see how options grow. In terms of hamburgers, each condiment offers a "yes" (include) or "no" (exclude) choice.
This is why the number of combinations is calculated using the power of 2 for each choice, resulting in a plethora of possibilities just from simple decisions.

Exponentiation

Exponentiation is a mathematical operation where a number is raised to the power of another number.
In simpler terms, it tells us how many times to multiply a number by itself. In the hamburger condiments scenario, exponentiation helps calculate the vast number of combinations.

• When you have 8 binary choices (one for each condiment), the number of combinations you can create is expressed as an exponent.
• Specifically, this is written as \(2^8\), meaning 2 multiplied by itself 8 times.

This concept is powerful because it translates simple choices into exponential growth. Using binary choice and exponentiation, we calculate the enormous quantity of potential hamburger configurations possible with just 8 condiments.

Combinatorial Enumeration

Combinatorial enumeration is the process of counting or listing all possible combinations in a set. It plays a crucial role in solving problems involving arrangements and combinations.
In our hamburger example, combinatorial enumeration is used to tally all the condiment combinations.

• Each condiment represents a variable in our set with two states: included or not.
• Determining the total number of combinations is essentially counting all possible ways to arrange these binary choices.

By enumerating the combinations systematically, we use the principles of combinatorics to ensure all possibilities are considered.
It's a handy technique not just in everyday problems but in advanced mathematical and scientific research as well.

Mathematical Reasoning

Mathematical reasoning helps us solve problems logically and systematically. It involves the use of logic to link together the mathematical facts we know to arrive at a solution.

• The hamburger condiment problem displays mathematical reasoning by structuring a logical sequence of steps.
• We start by acknowledging each condiment as a binary choice, understanding the total combinations through exponentiation, and then finally calculating the product.

This reasoning process is core in problem-solving, ensuring that each step logically follows and supports the next.
By laying out the choices, applying exponentiation, and confirming with multiplication, mathematical reasoning validates the solution and offers a clearer understanding of how numbers interact in different scenarios.