Permutations
Summary
TLDRThis 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
🔢 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.
📐 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.
Mindmap
Keywords
💡Permutations
💡Ordered arrangement
💡Distinct objects
💡Length of permutation
💡Basic counting principle
💡Factorial
💡Permutation rule
💡nPr
💡Special permutation rule
💡Combinatorics
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.
Посмотреть больше похожих видео
5.0 / 5 (0 votes)