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Chapter 5: Problem 15

Find the value of each permutation. $$ { }_{5} P_{0} $$

Short Answer

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Step by step solution

01

- Understand Permutation Notation

The notation \(_{n}P_{r}\) represents the number of permutations of n items taken r at a time. It is calculated using the formula \(_{n}P_{r} = \frac{n!}{(n-r)!}\).
02

- Apply the Given Values

Here, n = 5 and r = 0. Substitute these values into the permutation formula: \( _{5}P_{0} = \frac{5!}{(5-0)!} \)
03

- Calculate Factorials

Calculate the factorials. \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \) and \( 5! = 120 \) as well because \(5 - 0 = 5\).
04

- Solve the Division

Substitute the factorial values back into the formula: \( _{5}P_{0} = \frac{120}{120} \).
05

- Simplify the Expression

Simplify the expression: \( _{5}P_{0} = 1 \).

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Key Concepts

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

Permutation Formula
Permutations are a fundamental concept in combinatorics. They refer to the different ways you can arrange a set of items. The formula for permutations is given by \(_{n}P_{r} = \frac{n!}{(n-r)!}\). This equation helps determine the number of ways to arrange \('n'\) items taking \('r'\) items at a time. The exclamation mark \( ! \) represents the factorial operation, which we will cover next. Make sure you understand this formula deeply as it's essential in solving permutation problems.
Factorials
Factorials are crucial in the computation of permutations. The factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \). It is denoted by \( n! \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow very rapidly with increasing \( n \), making them an interesting yet computationally intensive concept. In the given exercise, you compute \( 5! = 120 \). Also, since \( 5 - 0 = 5 \), we realize \( 5! \) appears twice in the formula, simplifying our calculations.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations and permutations. It's about counting, arranging, and optimizing. One key area within combinatorics is studying permutations鈥攗nderstanding how to compute the number of ways to arrange items. For instance, suppose you have 5 different books and want to see in how many ways you can pick and arrange them. By applying the permutation formula \(_{5}P_{0} = \frac{5!}{(5-0)!} = \frac{120}{120} = 1\), you determine that there's only 1 way to arrange them鈥攅ssentially doing nothing, which makes sense mathematically.
Problem Solving Steps
Solving permutation problems can be broken down into clear steps:

1. **Understand Permutation Notation**: Know that \(_{n}P_{r}\) represents permutations of \( n \) items taken \( r \) at a time.
2. **Apply Given Values**: Substitute the given values into the permutation formula.
3. **Calculate Factorials**: Compute the necessary factorials.
4. **Solve the Division**: Divide the factorial results based on the formula.
5. **Simplify the Expression**: Simplify to find the final permutation value.

By following these steps, you systematically and accurately solve permutation questions, making the process easier and more understandable.

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Most popular questions from this chapter

According to the U.S. National Center for Health Statistics, \(0.15 \%\) of deaths in the United States are 25 - to 34-year-olds whose cause of death is cancer. In addition, \(1.71 \%\) of all those who die are \(25-34\) years old. What is the probability that a randomly selected death is the result of cancer if the individual is known to have been \(25-34\) years old?

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