Problem 53 - A Graphical Approach to College Algebra

Question

\text {Solve each problem involving combinations.} Hamburger Choices 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: Identify the Elements

First, identify the options you have. Howard's Hamburger Heaven offers the following 6 extras for hamburgers: cheese, relish, lettuce, tomato, mustard, and ketchup. We need to choose any 3 of these extras.

Step 2: Understand Combinations

Realize that this problem involves combinations, not permutations, because the order in which the extras are added does not matter. Therefore, we use the combination formula.

Step 3: Recall the Combination Formula

The formula for combinations is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items, and \( r \) is the number of items to choose.

Step 4: Substitute Values Into the Formula

Substitute \( n = 6 \) and \( r = 3 \) into the combination formula: \[ C(6, 3) = \frac{6!}{3!(6-3)!} \]

Step 5: Calculate Factorials

Calculate the factorials needed: \[ 6! = 720 \]\[ 3! = 6 \]\[ (6-3)! = 3! = 6 \]

Step 6: Solve the Formula

Plug the factorial values into the formula:\[ C(6, 3) = \frac{720}{6 \times 6} = \frac{720}{36} = 20 \]

Step 7: Conclusion

There are 20 different combinations of hamburgers that can be made with any 3 of the 6 extras.

Key Concepts

Factorials

A factorial is a mathematical operation that is applied to a whole number. It's denoted by an exclamation mark (!). For example, the factorial of 6, written as 6!, is calculated by multiplying all whole numbers from the given number down to 1:

• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

This operation is pivotal when solving combination problems because it helps us calculate the total number of possible arrangements of items. Factorials decrease rapidly as the number increases, indicating steep growth in potential arrangements.
Factorials are foundational in probability and statistics as they provide a way to count the possible permutations of a set. Understanding how to calculate a factorial is crucial to effectively applying the combination formula.

Combination Formula

The combination formula is a way to calculate how many different groups can be made from a larger set, where the order of selection does not matter. This is different from permutations where order does count. The formula for combinations is expressed as: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] Here, \( n \) stands for the total number of items you have to choose from, and \( r \) is the number of items you want to combine or select. What makes this formula powerful is the cancellation of unnecessary orders through dividing with the factorial of \( r \) and \( n-r \).
Let's apply this to the hamburger example: To find out how many ways you can select 3 extras out of 6, substitute \( n = 6 \) and \( r = 3 \) into the formula:

• \[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{720}{6 \times 6} = \frac{720}{36} = 20 \]

Thus, there are 20 different combinations possible.

Permutations

Permutations are similar to combinations, but with a crucial difference: order matters in permutations. If permutations were being used, each unique sequence of selected items would count as a different permutation.
The formula for permutations, when selecting \( r \) items from a set of \( n \), is:\[ P(n, r) = \frac{n!}{(n-r)!} \]In scenarios where you need to determine how many ways you can arrange a set of items, permutations are the tool to use. For example, if Howard's Hamburger Heaven wanted to know how many different sequences of 3 extras could be added to the hamburgers, permutations instead of combinations would be more appropriate.
However, in our problem, because the order of extras doesn't change the outcome (a hamburger's flavor won't change if you add lettuce before tomato), permutations aren't applicable. This is why we rely on combinations for this particular scenario.