Permutations

Summary

This script introduces permutations as ordered arrangements of distinct objects. It explains how to calculate the number of permutations for different lengths using the basic counting principle, leading to the concept of factorials. The script further discusses permutations of varying lengths from a set of letters, introduces the notation nPk for permutations of length k from n items, and applies these concepts to real-world scenarios like committee selection and art exhibitions.

Takeaways

• 🔑 A permutation is an ordered arrangement of distinct objects.
• 🔑 The number of permutations of length three from three distinct letters is calculated as 3 factorial (3!).
• 🔑 The number of permutations of length four from four distinct letters is calculated as 4 factorial (4!).
• 🔑 The general formula for permutations of length n from n distinct letters is n factorial (n!).
• 🔑 For permutations of length k from n distinct letters, the formula is nPk, which is the product of n down to (n-k+1).
• 🔑 The permutation rule, nPk, is used to calculate the number of ways to choose k items from n distinct items where order matters.
• 🔑 When the length of the permutation equals the number of available letters, the formula simplifies to n factorial (n!).
• 🔑 The special permutation rule is used when the number of items to arrange is equal to the number of available items.
• 🔑 Permutations are a fundamental concept in combinatorics, useful for problems involving ordered selections.
• 🔑 Practical applications of permutations include selecting committee members with distinct roles or arranging items in order, such as pieces of art for an exhibition.

Q & A

What is a permutation?

A permutation is an ordered arrangement of distinct objects.

How many permutations of length three can be made from three distinct letters?

There are 3 factorial (3!) permutations of length three from three distinct letters.

What is the total number of permutations of length four from four distinct letters?

There are 4 factorial (4!) permutations of length four from four distinct letters.

What is the general formula for the number of permutations of length n from n distinct letters?

The total number of permutations is n factorial (n!).

How many permutations of length two can be made from five distinct letters?

There are 5 times 4, or 20, permutations of length two from five distinct letters.

How many permutations of length three can be made from five distinct letters?

There are 5 times 4 times 3, or 60, permutations of length three from five distinct letters.

What does the notation nPk represent?

nPk represents the number of permutations of length k from n distinct letters.

What is the formula for calculating nPk?

The formula for calculating nPk is n * (n - 1) * (n - 2) * ... * (n - k + 1).

What is the special permutation rule when the length of a permutation is the same as the number of available letters?

The special permutation rule is nPn, which is equivalent to n factorial (n!).

How many ways are there to select three people out of six for a committee with distinct roles?

There are 6P3 ways to select three people out of six for a committee with distinct roles.

How many ways are there to select five pieces of art out of nine for an exhibition where the order matters?

There are 9P5 ways to select five pieces of art out of nine for an exhibition where the order matters.

Outlines

00:00

🔢 Understanding Permutations

This paragraph introduces the concept of permutations, which are ordered arrangements of distinct objects. It explains that permutations can be of different lengths and provides examples using letters and numbers. The paragraph also discusses how to calculate the total number of permutations for a given set of objects. It explains that the number of permutations is determined by the factorial of the number of objects when the length of the permutation equals the number of objects. For permutations of different lengths from a larger set, the calculation involves multiplying the number of objects by the number of choices remaining at each step, decreasing by one. The paragraph concludes with the introduction of the permutation formula, denoted as nPk, which is used to calculate the number of permutations of length k from n distinct objects.

05:01

📐 Applying Permutation Rules

The second paragraph delves into the application of permutation rules. It explains how to use the permutation formula to find the number of permutations of a certain length from a set of objects. The paragraph provides examples of calculating permutations for different lengths and object sets, such as 5P2 and 10P4. It also introduces the special permutation rule for when the length of the permutation is equal to the number of available objects, simplifying the formula to the factorial of the number of objects. The paragraph concludes by applying these rules to practical scenarios, such as selecting committee members or pieces of art for an exhibition, where the order of selection matters, and emphasizes the importance of permutations in combinatorics.

Keywords

💡Permutations

Permutations are the various ways in which a set of items can be arranged in a specific order. In the context of the video, permutations are used to describe the ordered arrangements of distinct objects. For example, 'THR' and 'HRT' are permutations of the letters T, H, and R, but they are not the same due to the different order, which is crucial in permutations. The video uses permutations to illustrate how many different ways items can be ordered, which is a fundamental concept in combinatorics.

💡Ordered arrangement

An ordered arrangement refers to the sequence in which items are placed. The video emphasizes that the order of elements in a permutation is significant, as changing the order results in a different permutation. For instance, '123' and '231' are distinct permutations of the numbers 1, 2, and 3 because the elements are arranged differently.

💡Distinct objects

Distinct objects in permutations are items that are unique and non-repetitive. The video script mentions that permutations are created from distinct objects, meaning no object is repeated within a single permutation. This is evident when discussing permutations like 'THR' from the letters T, H, and R, where each letter is used only once.

💡Length of permutation

The length of a permutation refers to the number of items in the arrangement. The video explains that the length is an important factor in calculating the total number of permutations possible. For example, permutations of length three from three letters would involve choosing three letters consecutively, leading to a calculation of 3 factorial.

💡Basic counting principle

The basic counting principle is a fundamental concept in combinatorics that states if there are m ways to do one thing and n ways to do another, then there are m * n ways to do both. The video uses this principle to calculate permutations, showing that if you have n distinct items and you want to arrange k of them, you multiply the number of choices at each step, decreasing by one each time.

💡Factorial

A factorial, denoted by n!, is the product of all positive integers from 1 to n. In the video, factorials are used to calculate the total number of permutations when the number of items to arrange equals the number of available items. For example, the number of permutations of length three from three letters is calculated as 3 factorial.

💡Permutation rule

The permutation rule, symbolized as nPk, is a formula used to calculate the number of permutations of length k from a set of n distinct objects. The video introduces this rule as a general method to find permutations and applies it to various examples, such as calculating 5P2 for permutations of length two from five letters.

💡nPr

nPr (read as 'n permute r') is a notation used to represent the number of permutations of length r from a set of n items. The video uses this notation to denote the unknown quantity of permutations and to apply the permutation rule formula. It's a shorthand way of expressing the calculation of permutations.

💡Special permutation rule

The special permutation rule is a specific case of the permutation rule where the length of the permutation is the same as the number of available letters (n = k). The video mentions that in such cases, you replace k with n in the formula to obtain the number of permutations, which is n factorial.

💡Combinatorics

Combinatorics is an area of mathematics concerned with counting, arrangement, and combination of sets and elements. The video discusses permutations as one of the building blocks of combinatorics, highlighting its importance in understanding various counting problems, such as selecting committee members or arranging art pieces.

Highlights

Definition of permutation as an ordered arrangement of distinct objects.

Example of permutations using letters T, H, and R.

Explanation of why THR and HRT are different permutations.

Example of permutations using numbers 1, 2, and 3.

Calculation of permutations of length three from three letters using three factorial.

Calculation of permutations of length four from four letters using four factorial.

General formula for permutations of length n from n letters using n factorial.

Calculation of permutations of length two from five letters using 5 times 4.

Calculation of permutations of length three from five letters using 5 times 4 times 3.

General formula for permutations of length k from n letters using the product n*(n-1)*...*(n-k+1).

Introduction of the notation nPk for permutations of length k from n letters.

Application of the permutation rule to calculate 5P2.

Application of the permutation rule to calculate 10P4.

Special permutation rule when the length of permutation equals the number of available letters.

Calculation of permutations of lengths four from four letters using 4 factorial.

Calculation of permutations of lengths six from six letters using 6 factorial.

Application of permutations to select committee members with distinct roles.

Application of permutations to select art pieces for an exhibition where order matters.

Explanation of how to calculate permutations for selecting three people out of six or five pieces of art out of nine.

Discussion of permutations as a fundamental concept in combinatorics.

Transcripts

Next, we will discuss the idea of permutations -

what are they and how to work with them.

Let's start with the definition. An ordered

arrangement of distinct objects is called a

permutation.

For example, THR is a permutation of length

three from letters T, H, and R. HRT is a permutation

of length three from letters T, H, and R. HR is a

permutation of length two from letters T, H, and R. Note

that permutation THR and permutation HRT are

not the same. Here's another example.

123 is a permutation of length three from

letters one, two, and three. 231 is a

different permutation of length three from the

same letters one, two, and three. 31 is a

permutation of length two from letters one, two,

and three.

How many permutations of lengths three out of

three letters are there? To create a permutation

we have to consecutively choose three letters.

Since the letters are all distinct the number of

choices that each step is decreasing by one so, by

the basic counting principle, the total number of

permutations is three times two times one which is

three factorial. How many permutations of

length four out of four letters are there? To

create the permutation we have to consecutively

choose four letters and since the letters are all

distinct the number of choices of each step is

decreasing by one. So by the basic cutting

principle, the total number of permutations is four

times three times two times one which is four

factorial. In general, how many permutations of

length (n) out of (n) letters are there? To create

a permutation, we have to consecutively choose

(n) letters and since the letters are all

distinct the number of choices at each step is

decreasing by one. So by the basic counting

principle, the total number of permutations is (n)

factorial.

How many permutations of lengths two out of five

letters are there? To create a permutation we have

to consecutively choose two letters. Since the

letters are all distinct the number of choices at

each step is decreasing by one. So by the basic

counting principle, the total number of permutations

is five times four.

How many permutation of length three out of

five letters are there?

To create a permutation, we have to consecutively

choose three letters. And since the letters are

all distinct the number of choices at each step is

decreasing by one. So by the basic counting

principle, the total number of permutations is five

times four times three. Finally, how many

permutations of length (k) out of (n) letters are

there?

To create a permutation, we have to consecutively

choose (k) letters. Since the letters are all

distinct the number of choices at each step is

decreasing by one. So by the basic counting

principle, the total number of permutations is

given by the following product.

In general, we want to find out how many

permutations of length (k) out of (n) letters are there.

Let's denote this unknown quantity using the

following symbols -nPk. Based on the observed

pattern, we obtain the following formula called the

permutation rule.

Let's do a few applications of this rule. How many

permutations of length two out of five letters

are there? The answer is 5P2 which we now can

compute using the formula. How many permutations

of lengths 4 out of 10 letters are there? The

answer can be expressed as 10P4 and can now

be computed using the formula.

If the length of a permutation is the same as the

number of available letters then we replace

(k) with (n) in the formula and obtain the

special permutation rule.

Let's do a few applications of the special

permutation rule. How many permutations of

lengths four out of four letters are there? The

answer is 4P4 which is 4 factorial. How many

permutations of lengths six out of six letters

are there? The answer is 6P6 which is six

factorial.

The following questions: "How many ways are there

to select three people out of six for a committee

with distinct roles?" and "How many ways are there

to select three pieces of art out of six for an

exhibition in which the order matters?" along with

many other questions have the same answer.

6P3

The following questions: "How many ways are there

to select five people out of nine for a committee

with distinct roles?" and "How many ways are there

to select five pieces of art out of nine for an

exhibition in which to order matters?" along with

many other questions have the same answer.

9P5

We discussed the idea of permutations which is

one of the building blocks of the combinatorics.