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Chapter 2: Problem 16

An order for a computer system can specify memory of 4 \(8,\) or 12 gigabytes and disk storage of \(200,300,\) or 400 gigabytes. Describe the set of possible orders.

Short Answer

Expert verified
There are 9 possible orders, combining each memory size with each disk size.

Step by step solution

01

Identify Specifications

Understand that there are two specifications to consider in the order: memory size and disk storage size.
02

List Memory Options

The memory options available are 4 GB, 8 GB, and 12 GB.
03

List Disk Storage Options

The disk storage options available are 200 GB, 300 GB, and 400 GB.
04

Combine Options

Combine each memory option with each disk storage option to determine all possible orders.
05

Enumerate Combinations

Enumerate the set of possible orders: 1. (4 GB, 200 GB) 2. (4 GB, 300 GB) 3. (4 GB, 400 GB) 4. (8 GB, 200 GB) 5. (8 GB, 300 GB) 6. (8 GB, 400 GB) 7. (12 GB, 200 GB) 8. (12 GB, 300 GB) 9. (12 GB, 400 GB)

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Key Concepts

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

Set Enumeration
When solving problems like determining possible computer system orders, set enumeration is a vital technique. Set enumeration involves listing all possible combinations within a given set. Here, we deal with combinations of memory sizes and disk storage options.
  • Start by identifying the separate categories or sets in your problem.
  • List all possible values within each set, considering them individually first.
  • Combine these lists to create a complete set of possible outcomes, ensuring no combinations are left out.
By using set enumeration, you thoroughly account for every feasible configuration, which in our example, are the nine specific combinations of memory and disk options.
Discrete Mathematics
Discrete Mathematics deals with countable, distinct elements. It is particularly useful in computer science, cryptography, and logic. In this context, we're examining combinations of computer specifications which are finite and distinct.
  • Focus on individual objects (such as data or numbers) and their combinatorial aspects.
  • Use principles and structures like sets, graphs, and logical statements to solve problems involving discrete elements.
By applying discrete mathematics principles, you gain insights into designing algorithms and solving complex problems through logical reasoning and structured approaches.
Product Rule
The product rule is a fundamental principle in combinatorics used to determine the number of possible combinations. When trying to find all possible orders for computer systems, the product rule states that if there are \(n\) ways to do one thing and \(m\) ways to do another, there are \(n \times m\) total possibilities.
  • Apply the rule by multiplying the number of choices available in each independent category.
  • In our example, multiply the three memory options by the three disk storage options; \(3 \times 3 = 9\).
This simple yet powerful rule swiftly guides you to the total number of combinations, ensuring no potential order is overlooked.
Combination Theory
Combination theory explores how to select items from a larger pool and forms a backbone for many combinatorial problems in discrete mathematics. The computer system order example uses a basic concept from combination theory, where you're selecting one element from two different sets (memory and disk).
  • Combination theory can involve more complex scenarios, like choosing subsets or handling constraints.
  • It helps to understand how elements interact across different sets, which is invaluable for optimization and probability calculations.
By mastering combination theory, you become adept at designing solutions that require selecting and arranging different elements effectively.

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