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Take # of Subjects 20! (20-5)! divided by Using # of subjects in pool by # Subjects selected Use n! = nx/n-1)! to expand the expression 20x 19 x 18 X 17X16X15! 20-56 (151) Subtract 20X19X18X17X16X15! 15! C

Take # of Subjects 20! (20-5)! divided by Using # of subjects in pool by # Subjects selected Use n! = nx/n-1)! to expand the expression 20x 19 x 18 X 17X16X15! 20-56 (151) Subtract 20X19X18X17X16X15! 15! C

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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The first question I have the answer.

The second question is: explain how this compares with 20C5 and 20P5.

**Calculating Combinations: Step-by-Step Process** To find the number of ways to choose a subset of subjects from a larger group, use combinations and factorials. Here's a walkthrough of the process: 1. **Identify the Total Number of Subjects and Selection Size:** - You have 20 subjects and want to select 5. 2. **Set up the Basic Combination Formula:** - Use the formula \(\frac{n!}{(n-r)!}\) where \(n\) is the total number of subjects (20) and \(r\) is the number of subjects you want to select (5). - This becomes \(\frac{20!}{(20-5)!}\). 3. **Simplify the Factorial Expression:** - \(20! = 20 \times 19 \times 18 \times 17 \times 16 \times 15!\) - \((20-5)! = 15!\) 4. **Expand and Simplify:** - Write the expanded form: \(20 \times 19 \times 18 \times 17 \times 16 \times 15!\) - Since \(15!\) appears in both the numerator and denominator, you can cancel it out. 5. **Perform the Multiplication:** - Calculate \(20 \times 19 \times 18 \times 17 \times 16 = 1,860,480\). - This is the number of ways to select 5 subjects out of 20. 6. **Result:** - The final answer is 1,860,480. This approach illustrates the simplification of factorial expressions in combinations using cancellation, resulting in an easier computation.
Transcribed Image Text:**Calculating Combinations: Step-by-Step Process** To find the number of ways to choose a subset of subjects from a larger group, use combinations and factorials. Here's a walkthrough of the process: 1. **Identify the Total Number of Subjects and Selection Size:** - You have 20 subjects and want to select 5. 2. **Set up the Basic Combination Formula:** - Use the formula \(\frac{n!}{(n-r)!}\) where \(n\) is the total number of subjects (20) and \(r\) is the number of subjects you want to select (5). - This becomes \(\frac{20!}{(20-5)!}\). 3. **Simplify the Factorial Expression:** - \(20! = 20 \times 19 \times 18 \times 17 \times 16 \times 15!\) - \((20-5)! = 15!\) 4. **Expand and Simplify:** - Write the expanded form: \(20 \times 19 \times 18 \times 17 \times 16 \times 15!\) - Since \(15!\) appears in both the numerator and denominator, you can cancel it out. 5. **Perform the Multiplication:** - Calculate \(20 \times 19 \times 18 \times 17 \times 16 = 1,860,480\). - This is the number of ways to select 5 subjects out of 20. 6. **Result:** - The final answer is 1,860,480. This approach illustrates the simplification of factorial expressions in combinations using cancellation, resulting in an easier computation.
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