Problem 28 - Algebra 2

Question

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities. selecting three of fifteen flavors of ice cream at the grocery store

Step-by-Step Solution

Answer

This situation is a combination with 455 possible selections.

Step 1: Understanding the Situation

The task is to select three flavors of ice cream out of fifteen available flavors. We need to determine whether this selection involves a permutation or a combination.

Step 2: Determine Permutation or Combination

Since the order in which we select the ice cream flavors does not matter, this is a combination problem. Each flavor selection is equal, regardless of the order they are picked.

Step 3: Apply Combination Formula

To find the number of combinations, we use the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]Here, \(n = 15\) (total flavors) and \(r = 3\) (flavors to select).

Step 4: Calculate the Combination

Substitute the values into the formula:\[ C(15, 3) = \frac{15!}{3!(15-3)!} = \frac{15!}{3!\times 12!} \]Simplify step by step:\[ 15 \times 14 \times 13 \div (3 \times 2 \times 1) = 5 \times 7 \times 13 = 455 \]

Step 5: Conclusion

The total number of ways to select three flavors of ice cream from fifteen is 455. This confirms the situation involves a combination.

Key Concepts

Permutations

When exploring arrangements, the term "permutations" often comes into play. Permutations are all about the order of items. If you change the order, you create a different permutation. For example, if you're lining up books on a shelf, changing the order alters the look of the shelf. This is a classic example where permutations apply.

Permutations are calculated when the arrangement or sequence matters. The permutation formula is used in these contexts, and it's defined as:\[ P(n, r) = \frac{n!}{(n-r)!} \]

Here, \(n\) is the total number of items, and \(r\) is the number of items you want to arrange. This concept mostly appears in scenarios like organizing books, assigning seats, or ordering tasks where each sequence is unique.

Factorial

Factorial is a mathematical operator, denoted by an exclamation point ("). It is used to calculate permutations and combinations. Factorial represents the product of an integer and all the integers below it. For example, \(5!\) is equivalent to \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials are a fundamental part of calculating permutations and combinations. They handle the counting of different arrangements or selections possible.

• In permutations, factorials help account for the different possible arrangements of \(r\) items within \(n\) options.
• In combinations, factorials help reduce the number of arrangements by dividing out sequences where the order doesn't matter.

Understanding factorials is key to mastering more complex counting methods utilized in various problems, including selection problems.

Selection Problems

Selection problems involve choosing items from a larger set. These problems can either be permutations or combinations. The primary difference is whether the order of selection matters.

• Permutation Selection: Useful when each sequence is unique and order matters.
• Combination Selection: Applied when just the choice itself matters, without regard to the order.

In the ice cream example, selecting three flavors from fifteen, order is irrelevant. Hence, it becomes a combination problem. Figuring out whether a selection is a permutation or combination helps apply the correct formula, ensuring accurate solutions.

By breaking down the problem into steps, you can confidently solve a wide range of selection problems.

Combination Formula

In scenarios where order does not matter, we make use of the combination formula. This formula helps calculate how many possible groups can be formed. The combination formula is written as follows:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Here, \(n\) represents the total number of items, and \(r\) is the number of items you wish to select. Let's go back to our ice cream problem. You have 15 flavors and wish to select 3, making \(n = 15\) and \(r = 3\).

Using the combination formula lets you calculate the total possible selections (combinations) with these parameters. It's essential to substitute the values into the formula correctly and simplify to reach the final answer.

The importance of applying this correctly becomes clear when handling real-world situations that involve making selections without concern for order, like team-member selections or food choices as in our example.