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Chapter 2: Problem 36

An assembly operation in a manufacturing plant requires three steps that can be performed in any sequence. How many different ways can the assembly be performed?

Short Answer

Expert verified
There are 6 different ways to perform the assembly.

Step by step solution

01

Understanding the Problem

We need to determine how many different sequences can be made using three steps. This is a classic permutation problem where we want to arrange a set of distinct items.
02

Identifying the Formula

Since we have three distinct steps, the number of ways to arrange these steps is given by the formula for permutations of distinct items, which is the factorial of the number of items, denoted as \(n!\). Here, \(n = 3\).
03

Calculating the Factorial

To find the number of different sequences, calculate \(3!\) (3 factorial), which means multiplying all positive integers up to 3, i.e., \(3 \times 2 \times 1\).
04

Solving the Calculation

Calculate \(3 \times 2 = 6\) and then multiply by 1. Thus, \(3! = 6\).
05

Conclusion

Therefore, there are 6 different ways the assembly operation can be performed in the given scenario.

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Key Concepts

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

Understanding Factorials
Factorial is a fundamental concept in mathematics, particularly in permutations and combinations. It is denoted by an exclamation mark (!). The factorial of a positive integer \( n \) is the product of all positive integers up to \( n \). For instance, \( 3! \) is calculated as \( 3 \times 2 \times 1 = 6 \).

Factorials are used to calculate the number of ways to arrange or sequence a set of objects. This is why they are crucial in problems requiring the arrangement of objects or steps in a sequence.
  • \( 3! = 6 \) implies there are 6 different ways to sequence 3 distinct objects.
  • Factorials grow incredibly fast, with \( 5! = 120 \), \( 10! = 3,628,800 \), etc.
Understanding factorials helps simplify complex permutation calculations and is a stepping stone for learning more about probability and statistics.
Sequence Arrangement in Permutations
When you need to determine how many different ways a set of items can be arranged, you are dealing with a sequence arrangement problem in permutations. Permutations consider the order in which items are arranged. This means that even if the same items are used, changing their order results in different permutations.

For example, in a manufacturing process with three assembly steps (A, B, and C), the different permutations would include:
  • ABC
  • ACB
  • BAC
  • BCA
  • CAB
  • CBA
Each sequence represents a unique way to perform the tasks. If there are \( n \) items to arrange, the number of permutations is calculated as \( n! \), highlighting how critical understanding factorials is in solving these problems.
Permutations in Manufacturing Processes
In the context of a manufacturing process, permutations are particularly useful for understanding and improving the workflow of tasks or operations. When a process involves multiple stages or steps that can be performed in various orders, determining the best sequence can optimize efficiency and productivity.

Permutations allow companies to explore all potential sequences to identify:
  • Which order minimizes time or resource usage.
  • How different sequences affect overall production output.
  • Possible bottlenecks in manufacturing sequences.
In our initial example with three steps, calculating the permutations with the factorial function \( 3! = 6 \) helps visualize all potential sequences. This exploration aids in strategic planning and operational improvements within manufacturing plants.

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Most popular questions from this chapter

An accident victim will die unless in the next 10 minutes he receives some type A, Rh-positive blood, which can be supplied by a single donor. The hospital requires 2 minutes to type a prospective donor's blood and 2 minutes to complete the transfer of blood. Many untyped donors are available, and \(40 \%\) of them have type \(A\), Rh-positive blood. What is the probability that the accident victim will be saved if only one blood-typing kit is available? Assume that the typing kit is reusable but can process only one donor at a time.

If \(A\) and \(B\) are independent events with \(P(A)=.5\) and \(P(B)=.2\), find the following: a. \(P(A \cup B)\) b. \(P(\bar{A} \cap \bar{B})\) c. \(P(\bar{A} \cup \bar{B})\)

Suppose that \(A\) and \(B\) are two events such that \(P(A)+P(B)>1\). a. What is the smallest possible value for \(P(A \cap B) ?\) b. What is the largest possible value for \(P(A \cap B) ?\)

A retailer sells only two styles of stereo consoles, and experience shows that these are in equal demand. Four customers in succession come into the store to order stereos. The retailer is interested in their preferences. a. List the possibilities for preference arrangements among the four customers (that is, list the sample space). b. Assign probabilities to the sample points. c. Let \(A\) denote the event that all four customers prefer the same style. Find \(P(A)\).

The symmetric difference between two events \(A\) and \(B\) is the set of all sample points that are in exactly one of the sets and is often denoted \(A \Delta B\). Note that \(A \Delta B=(A \cap B) \cup(\bar{A} \cap B)\). Prove that \(P(A \Delta B)=P(A)+P(B)-2 P(A \cap B)\)
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