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Chapter 10: Problem 1

At a restaurant, you select three different side dishes from eight possibilities. Is this situation a permutation, a combination, or neither? Explain.

Short Answer

Expert verified
This is a combination, as the order does not matter.

Step by step solution

01

Understand the Scenario

We are choosing three different side dishes from a total of eight possible side dishes at a restaurant. The order in which we select the dishes does not matter, only the group of dishes chosen is important.
02

Define Permutations and Combinations

A permutation is an arrangement of items where the order matters. In contrast, a combination is a selection of items where the order does not matter.
03

Identify the Type of Selection

Since the order in which the side dishes are selected does not affect the grouping, this is a situation involving combinations.
04

Conclude

Given that the order does not matter in the selection of the side dishes, this is a combination.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutations
When we talk about permutations, we refer to arrangements where the order of the items is crucial. Imagine having a set of objects, such as colored balls, and forming different sequences with them. Each unique sequence counts as a different permutation. For instance, say you have three balls: red, blue, and green. The arrangement "red, blue, green" is different from "green, red, blue" because the order differs. Therefore, each sequence represents a unique permutation. In general, permutations are used in scenarios where the exact order in which things happen or get placed, matters. Thus, anytime the positioning or sequence makes a difference, we deal with permutations.
Order of Selection
Understanding the order of selection is crucial in distinguishing between permutations and combinations.
  • **Order Matters:** This is a typical sign that you are dealing with permutations. For example, choosing a leadership team where specific roles are assigned based on the order of selection requires permutations.
  • **Order Does Not Matter:** This signals that you're working with combinations. A common example is selecting a team without assigning specific roles; only the members in the group matter, not the order they were chosen in.
In the restaurant side dishes problem from the original exercise, the order of selection was mentioned not to affect the final grouping. Therefore, it highlights the concept of combinations rather than permutations. Understanding whether order matters can often simplify the approach and solution to similar mathematical problems.
Mathematical Problem Solving
Solving mathematical problems, especially those involving permutations and combinations, requires a clear understanding of the problem through step-by-step analysis. Here's a simple strategy to follow:
  • **Understand the Problem:** Break down the problem statement. Identify what's required and note key details.
  • **Define Key Terms:** Know the difference between permutations and combinations. Understand if the order is significant.
  • **Solve With Clarity:** Once you've understood the type of problem, apply relevant formulas or logic. Work through logically, showing each step clearly.
  • **Conclude with Confidence:** Based on the steps followed, make a clear conclusion. Ensure that the answer logically stems from your process.
For example, in the side dishes scenario, by understanding that order doesn鈥檛 matter, the problem was distilled into spotting a combination problem. This logical sequence not only helps in solving mathematical problems efficiently but boosts confidence in understanding complex problem scenarios.

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Most popular questions from this chapter

Create a tree diagram with probabilities showing outcomes when drawing two marbles with replacement from a bag containing one blue and two red marbles. (You do replace the first marble drawn from the bag before drawing the second.)

A chili recipe calls for seven ingredients: ground beef, onions, beans, tomatoes, peppers, chili powder, and salt. There are no directions about the order in which the ingredients should be combined. You decide to add the ingredients in a random order. a. How many different arrangements are there? b. What is the probability that onions are first? c. What is the probability that the order is exactly as listed above? d. What is the probability that the order isn't exactly as listed above? e. What is the probability that beans are third?

If you flip a paper cup into the air, what are the possible outcomes? Do you think the outcomes are equally likely? How can you test your conjecture?

A product like \(3 \cdot 2 \cdot 1\) or \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\) is called a factorial expression and is written with an exclamation point, like this: \(3 \cdot 2 \cdot 1=3\) ! and \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=5\) ! a. How can you calculate 8 !? b. How can you use factorial notation to calculate the number of permutations of 10 objects chosen 10 at a time? \([-\) See Calculator Note \(10 \mathrm{~F}\) to learn how to compute \(n\) with your calculator. \(\left.{ }^{-1}\right]\) c. Write an expression in factorial notation that can be used to calculate \({ }_{n} P_{n}\).

A teacher would like to use her calculator to randomly assign her 24 students to 6 groups of 4 students each. Create a calculator routine to do this.
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