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Magazine Chapter 5: Problem 15
Find the value of each permutation. $$_{5} P_{0}$$
Short Answer
Expert verified
The value is 1.
Step by step solution
01
Understanding the Permutation Notation
The notation \(_{n}P_{k}\) represents the number of permutations of \(n\) items taken \(k\) at a time. For this problem, \(_{5}P_{0}\), we need to find the permutations of 5 items taken 0 at a time.
02
Using the Permutation Formula
The formula for permutations is \(_{n}P_{k} = \frac{n!}{(n-k)!}\). Here, \(n = 5\) and \(k = 0\).
03
Calculating the Factorials
First, compute the factorial of 5: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Next, calculate the factorial of \((5-0)\) which is \(5! = 120\).
04
Substitute and Simplify
Substitute the computed factorials into the formula: \(_{5}P_{0} = \frac{120}{120} = 1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
factorials
A factorial is a fundamental concept in mathematics, especially in combinatorics. It's the product of all positive integers up to a given number. The factorial of a number \(n\) is denoted as \(n!\). For example:
Factorials take on an essential role in permutations and combinations, helping us understand various ways to organize and choose items. When you calculate a factorial, you are essentially figuring out how many different ways you can arrange a set of items.
- 0! = 1 (by definition)
- 1! = 1
- 2! = 2 \times 1 = 2
- 3! = 3 \times 2 \times 1 = 6
- 4! = 4 \times 3 \times 2 \times 1 = 24
- 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Factorials take on an essential role in permutations and combinations, helping us understand various ways to organize and choose items. When you calculate a factorial, you are essentially figuring out how many different ways you can arrange a set of items.
permutation formula
Permutations focus on the arrangement of items where the order matters. The formula to find the number of permutations of \(n\) items taken \(k\) at a time is given by: ewline ewline \[ _{n}P_{k} = \frac{n!}{(n-k)!} \] ewline For example, if you want to find the number of permutations of 5 items taken 0 at a time, you would use this formula where \( n = 5 \) and \( k = 0 \): ewline \[ _{5}P_{0} = \frac{5!}{(5-0)!} = \frac{120}{120} = 1 \] ewline This means there is exactly one way to arrange 5 items when you are taking 0 at a time. It might seem counterintuitive, but this is a mathematical rule reflecting the concept of doing nothing, which is possible in exactly one way.
combinatorial mathematics
Combinatorial mathematics is a branch of mathematics dealing with combinations, permutations, and the general study of counting. It's widely used in many fields, from computer science to biology. Here are some key concepts:
- Combinations: Ways of selecting items from a group where order doesn't matter.
- Permutations: Ways of arranging items from a group where order does matter.
- Factorials: The product of all positive integers up to a number \(n\).
understanding notation
Understanding mathematical notation is crucial for solving problems efficiently. Let's break down a few key notations used in permutations and combinations:
- \( n! \) (Factorial notation): Represents the factorial of \(n\).
- \( _{n}P_{k} \) (Permutation notation): Represents the number of ways to arrange \(n\) items taken \(k\) at a time.
- \( _{n}C_{k} \) (Combination notation): Represents the number of ways to select \(k\) items from \(n\) items without regard to order.
