Problem 67 - Basic Statistics for Business and Economics with Student

Question

Several years ago Wendy's Hamburgers advertised that there are 256 different ways to order your hamburger. You may choose to have, or omit, any combination of the following on your hamburger: mustard, ketchup, onion, pickle, tomato, relish mayonnaise, and lettuce. Is the advertisement correct? Show how you arrive at your answer.

Step-by-Step Solution

Answer

Yes, Wendy's is correct. There are 256 different ways to order a hamburger with 8 ingredients.

Step 1: Identify Ingredients

There are 8 different ingredients: mustard, ketchup, onion, pickle, tomato, relish, mayonnaise, and lettuce. Each ingredient can either be included or excluded in a hamburger.

Step 2: Binary Choice per Ingredient

For every ingredient, you have two choices: include it or exclude it. Since these are independent choices, we can multiply the options for each topping.

Step 3: Calculate Total Combinations

Using the formula for combinations where there are 2 choices (include or exclude) per ingredient:\[ 2^n \]where \( n \) is the number of ingredients.So, with 8 ingredients, the calculation becomes:\[ 2^8 \]

Step 4: Compute Power of Two

Calculate \( 2^8 \), which means multiplying 2 by itself 8 times:\[ 2^8 = 256 \]

Step 5: Conclusion

Since we calculated there are 256 possible combinations, the advertisement by Wendy's is correct. Each configuration of ingredients represents a different way to order a hamburger.

Key Concepts

Binary Choices

When it comes to building your perfect hamburger, binary choices play a crucial role. For each of the 8 ingredients, you have two options: to include it or leave it out. This brings us to the idea of binary choices. A binary choice means that for every decision or component involved, you have only two possible states—yes or no, on or off, include or exclude.
In the context of Wendy's hamburger options, each ingredient like mustard, ketchup, or onion acts as a single binary decision point. This type of choice-making mirrors the binary language of computers, where 0 represents 'exclude' and 1 represents 'include'.
Understanding binary choices helps simplify complex problems, making it easier to see all possible configurations. With 8 different decision points (ingredients), we can create a structured way to count all possible combinations.

Combinations

In combinatorics, a combination refers to the selection of items from a larger pool, where the order doesn't matter. With Wendy's hamburger, the combinations give us the different ways we can choose the available ingredients. Each combination is a unique set of included ingredients.
For Wendy's, since you can either include or exclude each of the 8 ingredients, the total number of combinations is determined by calculating how many unique selections of the ingredients can be made. This is where the formula for combinations of binary choices comes into play:

• Each ingredient has 2 possibilities (include or exclude).
• The total number of combinations across all ingredients is calculated as the product of these possibilities.

You use the combinations formula \[ 2^n \]where \( n \) is the number of ingredients. In this case, that's 8, leading us to \[ 2^8 \], which we will simplify in the next section.

Power of Two

The concept of the power of two shows us how to calculate the total number of combinations in Wendy's hamburger example. The formula for calculating combinations of binary choices is \[ 2^n \], where each number is a binary choice. Here, \( n \) represents the number of items or ingredients, which is 8.
To compute the power of two means exploring what happens when we multiply the number 2 by itself multiple times. For Wendy's hamburgers:

• Calculate \( 2^8 \), which means multiplying 2 by itself 8 times: \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \).
• This results in 256.

The power of two is fundamental in determining how many possible combinations of hamburger ingredients are available. In this case, it confirms that there are indeed 256 different ways to customize your hamburger, affirming Wendy's advertisement.