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Chapter 4: Problem 1

What is the main difference between a situation in which the use of the permutations rule is appropriate and one in which the use of the combinations rule is appropriate?

Short Answer

Expert verified
Permutations consider order; combinations do not.

Step by step solution

01

Understanding Permutations

Permutations are used when we are interested in the order of the selection. In other words, a permutation is a specific sequence of objects and the sequence itself is significant. For example, if we have three books A, B, and C, placing them in different orders results in different permutations (e.g., ABC, ACB, BAC, etc.).
02

Understanding Combinations

Combinations are used when the order of selection does not matter. In this case, we are simply choosing a subset of objects regardless of their sequence. Using the same example of three books, selecting any two books out of three books yields the same combination whether it's AB, BA, or CB. Thus, the order does not create a new set.
03

Determining the Difference

The main difference between permutations and combinations is whether the sequence or order of the elements within the set matters. If order matters, then it is a permutation problem. If order does not matter, then it is a combination problem.

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Key Concepts

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

Permutations
Permutations are important when we're interested in arrangements where the order matters. Imagine you're organizing books on a shelf. The order in which you place them is significant; for example, arranging the books as ABC is different from ACB or BAC. When calculating permutations, we consider each unique sequence as distinct. The formula for finding permutations of choosing r objects from a set of n is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] Where \(n!\) (n factorial) is the product of all positive integers up to \(n\). Permutations are useful in scenarios like establishing seating arrangements or when ranking contestants in a competition.
Combinations
Combinations focus on selections where order does not matter. This means choosing a group of items from a larger set, and the order of those items is irrelevant. For example, if you're picking two books from a set of three (like ABC), picking AB is the same as picking BA. They count as the same combination.The formula for determining combinations is:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Where again \(n!\) is the product of all positive integers up to \(n\), and \(r!\) accounts for the number of positions being chosen. Combinations are typically used in lottery number selections or forming teams or groups without distinguishing the members' roles.
Order of Selection
The concept of the 'Order of Selection' is crucial in distinguishing between permutations and combinations. Understanding whether or not the sequence in which items are selected is critical will determine the approach.
  • When the sequence matters, it's essential to use permutations because each unique order counts as a different arrangement.
  • Conversely, if the sequence doesn't matter, we choose combinations, as we're only interested in the group of items itself, not how they're arranged.
Order of selection applies to real-world situations like organizing schedules (where order is important) or simply choosing team members for a task without assigning roles based on their order of selection.

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Most popular questions from this chapter

(a) Explain why \(-0.41\) cannot be the probability of some event. (b) Explain why \(1.21\) cannot be the probability of some event. (c) Explain why \(120 \%\) cannot be the probability of some event. (d) Can the number \(0.56\) be the probability of an event? Explain.

In a sales effectiveness seminar, a group of sales representatives tried two approaches to selling a customer a new automobile: the aggressive approach and the passive approach. For 1160 customers, the following record was kept: $$\begin{array}{lllr} \hline & \text { Sale } & \text { No Sale } & \text { Row Total } \\ \hline \text { Aggressive } & 270 & 310 & 580 \\ \text { Passive } & 416 & 164 & 580 \\ \text { Column Total } & 686 & 474 & 1160 \\ \hline \end{array}$$ Suppose a customer is selected at random from the 1160 participating customers. Let us use the following notation for events: \(A=\) aggressive approach, \(P a=\) passive approach, \(S=\) sale, \(N=\) no sale. So, \(P(A)\) is the probability that an aggressive approach was used, and so on. (a) Compute \(P(S), P(S \mid A)\), and \(P(S \mid P a)\). (b) Are the events \(S=\) sale and \(P a=\) passive approach independent? Explain. (c) Compute \(P(A\) and \(S)\) and \(P(P a\) and \(S)\). (d) Compute \(P(N)\) and \(P(N \mid A)\). (e) Are the events \(N=\) no sale and \(A=\) aggressive approach independent? Explain. (f) Compute \(P(A\) or \(S)\).

You roll two fair dice, a green one and a red one. (a) What is the probability of getting a sum of 6 ? (b) What is the probability of getting a sum of 4 ? (c) What is the probability of getting a sum of 6 or 4? Are these outcomes mutually exclusive?

(a) Make a tree diagram to show all the possible sequences of answers for three multiple-choice questions, each with four possible responses. (b) Probability extension: Assuming that you are guessing the answers so that all outcomes listed in the tree are equally likely, what is the probability that you will guess the one sequence that contains all three correct answers?

USA Today gave the information shown in the table about ages of children receiving toys. The percentages represent all toys sold. What is the probability that a toy is purchased for someone (a) 6 years old or older? (b) 12 years old or younger? (c) between 6 and 12 years old? (d) between 3 and 9 years old? A child between 10 and 12 years old looks at this probability distribution and asks, "Why are people more likely to buy toys for kids older than I am (13 and over) than for kids in my age group (10-12)?" How would you respond? $$\begin{array}{l|c} \hline \text { Age (years) } & \text { Percentage of Toys } \\ \hline 2 \text { and under } & 15 \% \\ 3-5 & 22 \% \\ 6-9 & 270 / 6 \\ 10-12 & 14 \% \\ 13 \text { and over } & 22 \% \end{array}$$
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