PERMUTATION

[JeffM]

I do not want to try to explain Dr. Peterson's words; he is the best one to do so.

I do think that he is correct that the question you are asking is at least partly about English. And, like all natural languages, meaning is often dependent on context and sometimes ambiguous even in context.

But here I think the issue is that, although "repetition" has one usual meaning, not multiple meanings, that meaning always has a context, namely what is being repeated. Moreover, when we ask what is the number of a set, we mean the number of distinct items without repetition of whatever it is that distinguishes one member of the set from all the others.

Let's go back to your arrangement of the letters in the word "lie." If the object is to determine how many distinct three-letter sequences can be formed from the three letters L, I, and E, the answer is 27. If the object is to determine how many distinct three-letter sequences can be formed from those three letters without using any of the letters more than once, the answer is 6. If the object is to determine how many distinct three-letter sequences can be formed from those three letters such that no letter is used successively, the answer is 12.

Let's ask the same questions about the letters in the word "lee." The answers are 8 rather than 27, 0 rather than 6, and 2 rather than 12.

Using the word "repetition" to distinguish among those six questions is simply poor English writing. Counting problems cannot be reduced to a single algorithm; they require careful thought before applying the mechanics of the counting principles. But you cannot think carefully about poorly written questions because it is not clear what they even mean.

 

[Saumyojit]

If the object is to determine how many distinct three-letter sequences can be formed from the three letters L, I, and E, the answer is 27

In our textbooks the same question will be given like this : How many arrangements can be made from the word lee (1)if repetiton is allowed then 8 (2) if repetiton is not allowed then i think 3 {lee,eel,ele} . But u have said it zero that means for u repetion not allowed means as u have said "without using any of the letters more than once " but for me repetion not allowed means not including duplicate arrangments.

 

[Dr.Peterson]

Now we're talking about the English used in your textbook, which is more concrete than the original question, and therefore worth discussing. I suspect it is different from our language, and we need to be sure what the author actually means.

I would like to see an image of such an example, so we can see for ourselves how they are using the terms. If possible, I'd want to see them defining "repetition allowed or not allowed", and then a solved problem using those words. The issue is not what you think, or what I think, but what they are teaching.

To me, it is nonsense to ask "How many arrangements can be made from the word LEE if repetition is not allowed?" At best, the answer would be 2 (EL and LE). Sometimes people talk about "permutations with repetition" to mean permutations of a multiset (that is, the repetition is within the source); so your "How many arrangements can be made from the word LEE?" is inherently a problem with repetition. When the E is repeated in the source, you can't then forbid repetition.

So I am very curious to see the textbook actually asking for this, and how they explain it.

 

[Saumyojit]

@Dr.Peterson What i have understood
How many ways the word "lee" can be arranged ?-->3 (lee,ele,eel) -->(PERMUTATION of MULTISET)

How many ways the word "lee" can be arranged if letters repeat?-->8 (lee,eee,lll,lle,ell,ele,lel,eel)-->("permutation with repetition")

How many ways the word "lee" can be arranged if letters dont repeat? -->2 (le,el) -->("permutation without repetition")

To me, it is nonsense to ask "How many arrangements can be made from the word LEE if repetition is not allowed?"

thats how we get asked i dont knw why it is nonsense.

Sometimes people talk about "permutations with repetition" to mean permutations of a multiset (that is, the repetition is within the source); so your "How many arrangements can be made from the word LEE?" is inherently a problem with repetition.

Permutation with repetition is differnt from perm. of multiset. In Permutation with repetition one letter from the source can fill up all of the r places whereas Perm of a multiset we can use just that amt of letters as the way its mutliplcity is given in the source set to fill only that amt of r places . Demonstration given above 1st case of (lee)

So i disagree with {"How many arrangements can be made from the word LEE?" is inherently a problem with repetition}--> for me its inherently a problemn of a multiset and permutations with repetition are not the same with permutations of a multiset. In wiki they have also stated with different headings.

 

[Dr.Peterson]

I asked for a definition and an example direct from your book. At this point, I'm ignoring what you or I think, because this is a language problem, and only the language of the source (that is, how the author intends the words) matters.

As JeffM pointed out, "permutations with repetition" by itself does not clearly distinguish among various possibilities. I said long ago that in my experience the terminology in this area is inconsistent, and I've found it difficult to find information about particular kinds of problems because the same words are used with different meanings, in different contexts. I want to know how they are being used in your context.

If you're referring to Wikipedia or some other wiki, please give a specific reference and quote the words. But that is not necessarily what you are being taught, which is what I want to know.

Permutation with repetition is differnt from perm. of multiset. In Permutation with repetition one letter from the source can fill up all of the r places whereas Perm of a multiset we can use just that amt of letters as the way its mutliplcity is given in the source set to fill only that amt of r places . Demonstration given above 1st case of (lee)

So i disagree with {"How many arrangements can be made from the word LEE?" is inherently a problem with repetition}--> for me its inherently a problemn of a multiset and permutations with repetition are not the same with permutations of a multiset. In wiki they have also stated with different headings.

Here is my issue: If you are given LEE and asked for arrangements without repetition, how is that different from being given LE?

Again, I want to see a reputable source that poses a problem in that way.

 

[Saumyojit]

"permutations with repetition" by itself does not clearly distinguish among various possibilities

give me a eg where it does not clearly distinguish among various possibilities

If you're referring to Wikipedia or some other wiki, please give a specific reference and quote the words. But that is not necessarily what you are being taught, which is what I want to know.

I just said it because u told in post #23 permutation with repetition is same with permutations of a multiset. IT canot be same even in wiki it is also mentioned under two different headings and in different pararagraphs.When something is mentioned under two different headings and in different pararagraphs they are different.[https://en.wikipedia.org/wiki/Permutation#Permutations_with_repetition]

If you are given LEE and asked for arrangements without repetition, how is that different from being given LE?

yes u are right.
How many ways the word "lee" or "le" can be arranged if letters dont repeat? -->2 (le,el) -->("permutation without repetition")
How many ways the word "le" can be arranged ?--> (le,el) -->2-->(PERMUTATION of MULTISET)
first two cases for the word "le" are the same. As u can see. SO PERMUTATION of MULTISET = permutation without repetition for the word 'le'


How many ways the word "le" can be arranged if letters repeat? will give--> 4 arrangments -->{ll,ee,le,el} -->"permutation with repetition")

 

[Dr.Peterson]

I can't reply properly until you show me what your textbook actually says. As I have told you, I am only going to discuss this in that context. You are refusing to cooperate.

give me a eg where it does not clearly distinguish among various possibilities

I told you, this is what post #21 says. Read it. Then read it again until you see what he is saying.

I just said it because u told in post #23 permutation with repetition is same with permutations of a multiset. IT canot be same even in wiki it is also mentioned under two different headings and in different pararagraphs.When something is mentioned under two different headings and in different paragraphs they are different.[https://en.wikipedia.org/wiki/Permutation#Permutations_with_repetition]

I said nothing of the sort. What I said was that it is meaningless to talk about permutation of a multiset without repetition, so there is no sense in using the word in such a problem at all.

In terms of the Wikipedia page, do you not see that it is not what you have been talking about at all? Or that in the section that follows, about permutations of a multiset, they never mention with or without repetition. My point in bringing that page up, in part, was to show that their usage is not yours, so that your usage is not standard.

That is why I need to see your textbook's usage, so we can talk about what you claimed is what you have been taught. Until you do so, I will have to ignore you. You're just talking about your personal feelings, which is a waste of time.

 

[JeffM]

Dr. Peterson is answering better than I could. But the point that he is making is, to my mind, EXTREMELY important. What the real-world question means is not a mathematical question at all; it is a question of language. When you use mathematics to work on real-world questions, you ASK QUESTIONS if you are not absolutely certain what is being asked. You work to clarify what the natural language question actually means BEFORE you even try to do mathematics. I created a very prosperous career for myself by asking many questions and then writing answers in very clear English based on well defined mathematical problems. It is a tragedy that modern teaching methods do not encourage or even enable asking the the problem setter to clarify what the problem means

In many respects, what you are worrying about is a purely scholastic issue: what kind of answer does the person who posed the question expect when you cannot ask what was intended. For someone with the confidence to ask questions, this is seldom if ever an issue outside an academic setting. (Of course, many lack the self-confidence to ask questions, but I doubt that is a problem for you.) I do not say that it is impossible to ask ambiguous questions in the language of mathematics, but I guarantee that it is amazingly easy to ask confusing or even meaningless questions in any natural language.

 

[Saumyojit]

I can't reply properly until you show me what your textbook actually says

I saw and quoting from "My text book" says the same exact thing.This 3 questions are written exactly same in my textbook. And the answers are 3,8,2. below 3 questions are given in textbook same to same.

How many ways the word "lee" can be arranged ?-->3 (lee,ele,eel) -->(PERMUTATION of MULTISET)

How many ways the word "lee" can be arranged if letters repeat?-->8 (lee,eee,lll,lle,ell,ele,lel,eel)-->("permutation with repetition")

How many ways the word "lee" can be arranged if letters dont repeat? -->2 (le,el) ("permutation without repetition")
I got it .

post #21 says. Read it. Then read it again until you see what he is saying.

In post no 21 JeffM said

repetition has one usual meaning not multiple meanings

Now i got it , Repeation always refers to the elements within an arrangement when any 1 elemnt's multiplcity is more than one . It has nothing to do with excluding "duplicates of arrangments.

If the object is to determine how many distinct three-letter sequences can be formed from those three letters such that no letter is used successively,

The same question with a subtle difference in result can be written as "how many distinct three-letter sequences can be formed from those three letters such that letter can be used successively. "-->THEN IT becomes Permu with repetion

Using the word "repetition" to distinguish among those six questions is simply poor English writing

After knowing what actually repetion means, its not difficult for me to distinguish among those six questions. I can understand now.
I have read post 21 ..i know he has used two diff words and also understood all the six cases.

What I said was that it is meaningless to talk about permutation of a multiset without repetition

Yes this line I agree , the input letter of the source set (any 1 letter atleast ) has to have multiplcity greater than 1. SO permutation of a multiset without repetition is not possible
@Dr.Peterson @JeffM

 

[Dr.Peterson]

I saw and quoting from "My text book" says the same exact thing.This 3 questions are written exactly same in my textbook. And the answers are 3,8,2. below 3 questions are given in textbook same to same.

How many ways the word "lee" can be arranged ?-->3 (lee,ele,eel) -->(PERMUTATION of MULTISET)

How many ways the word "lee" can be arranged if letters repeat?-->8 (lee,eee,lll,lle,ell,ele,lel,eel)-->("permutation with repetition")

How many ways the word "lee" can be arranged if letters dont repeat? -->2 (le,el) ("permutation without repetition")

Do you see that the problems as quoted here do not use the terms "with or without repetition" as if that fully defined the problem? The precise wording is important: "if letters repeat" says what has to be said.

The last one, though, still seems odd to me. When we talk about arranging [the letters of] a word, that usually means all the letters. (That's why I object to "without repetition" in the case of a multiset.) If I wrote that problem, I would say something like "arranging a subset of the letters".

At least we can tell what is intended, which is the important thing.

 

[Saumyojit]

Do you see that the problems as quoted here do not use the terms "with or without repetition" as if that fully defined the problem? The precise wording is important: "if letters repeat" says what has to be said.

How many ways the word "lee" can be arranged if letters repeat? it inherently means permutation with reputation ie. we can use any one letter from the source and it can fill up all of the r places . whats the problemn if they did not use the term with repetion . They have said letters repeat which is the other way of saying it.

The last one, though, still seems odd to me

u are perhaps confusing between permu without repetion and permu of multiset . THe last one is permu without repetion .In permu without repetion it is opposite of permutation with reputation that is we cannot use one letter more than 1 in each arrangment.
Whereas permu of multiset means k-permutation of a multiset M is a sequence of length k of elements of M in which each element appears a number of times less than or equal to its multiplicity in M .
understand the last line. I have understood.

 

[Dr.Peterson]

How many ways the word "lee" can be arranged if letters repeat? it inherently means permutation with reputation ie. we can use any one letter from the source and it can fill up all of the r places . whats the problemn if they did not use the term with repetion . They have said letters repeat which is the other way of saying it.

As I understand it (and I'm very confused as I try to follow this thread), we've been trying to figure out what is meant by "with or without repetition". Your initial post got that all wrong.

Since your book doesn't use those terms (as you've quoted it), that is irrelevant. That's all I was saying in the bit you quoted: you are no longer asking about the meaning of "with repetition", so we can drop that topic. Their way of wording it tells you what they want.

u are perhaps confusing between permu without repetion and permu of multiset . THe last one is permu without repetion .In permu without repetion it is opposite of permutation with reputation that is we cannot use one letter more than 1 in each arrangment.
Whereas permu of multiset means k-permutation of a multiset M is a sequence of length k of elements of M in which each element appears a number of times less than or equal to its multiplicity in M .
understand the last line. I have understood.

Here I wasn't talking about words at all, but the problem itself. It merely seems odd to even ask the question, "How many ways can the word "lee" be arranged if letters don't repeat?" since the extra "e" seems unnecessary. If your book asks that question, fine. I just wouldn't ask it. I'd just say "le".

Can we drop this? I'm tired of trying to follow an argument about words that aren't even used in your book.

 

[Saumyojit]

you are no longer asking about the meaning of "with repetition"

Yes coz now i know the meaning of repetiton . So drop it.

If your book asks that question, fine. I just wouldn't ask it. I'd just say "le"

The book asked it using "lee" not"le" coz if they asked using le it would be easy to answer .But i get ur point

I'm tired of trying to follow an argument about words that aren't even used in your book

How many ways the word "lee" can be arranged ?
this question in the book was given under the subtopic permutation of n things taken all together not all different

How many ways the word "lee" can be arranged if letters repeat? this was given under the subtopic permu with repetiton

 

[Dr.Peterson]

I like the name of the first subtopic more than the other. "Permutation of n things taken all together, not all different" is very precise, and the last phrase is a way to refer to multisets without using the term. "Permutation with repetition" is vague. That may be the problem.

 

[Saumyojit]

"last phrase is a way to refer to multisets without using the term. "Permutation with repetition" is vague.

Last phrase or question is different from the question of permu of multisets. PLEASE UNDERSTAND
How many ways the word "lee" can be arranged if letters repeat?(permu with repititon) they are telling each letter of the given word may repeat any no of times to fill up all r places or less than r places .(r is 3 so each letter can be present zero times or 1 time or 2time or 3 time) in each arrangement. (lee,eee,lll,lle,ell,ele,lel,eel)

How many ways the word "lee" can be arranged? this is refering to permu of multisets --> (lee,ele,eel)
answers are different and questions are from diff concepts

 

[Dr.Peterson]

What I meant by "the last phrase" (NOT "last question") was the single phrase "not all different". That says that it ("lee" for example) is not a set, but a multiset -- not all letters in it are distinct. That's all I said. I was complimenting their way of naming what they were talking about there, namely the use of enough words to make it clear what is intended.

And that is what we have been recommending for some time: Don't try to describe a particular type of problem using only three words, because that only leads to confusion.

 

[Saumyojit]

How many ways can you choose a letter that will be used twice in the word?

Two, a or b.

Given that one letter is to be used twice in a three-letter word, how many positions can be chosen for the other letter?

Three, first position, second position, or third position.

3 * 2 = 6.

@JeffM
Yes this is also a quick method .
I never thought of it.
I have the option of choosing either a or b.
1+1=2 options with any ONE OF LETTER which I will repeat twice .
So now I have any one position left out of 3
for the other letter.
For each of two positions occupied I have 1 place left.
Event 1:
If aa is at (1,3) then b is at 2nd place (1*1) that is I have 1 way of selecting the position of aa happening simultaneously with the event of selecting the position of other letter that why multiplication in between
So this was one whole Event/one unique arrangement, then right?
Or(+)
This is another event below which will happen if other events does not happen.

If aa is at (1,2) then b is at 3rd(1*1).
THIS IS another way of selecting the position of aa simultaneously happening with choosing the position of b
Or
This (below) is also one mutually exclusive event which will happen if no other events happen.
Thats why we use addition in between every events.

If aa is at (2,3) then b is at 1st(1*1)

So For (aa)we see that there are 3 exclusive events (1*1+1*1+1*1)
Similarly for (BB) there will be 3 exclusive events
Either one of 3 aa event or one of 3 BB event will happen at a single time=(1*1+1*1+1*1+1*1+1*1+1*1)

6 events all total
This is what you are trying to say by "How many positions can be chosen for the other letter"
I see that we are not only choosing the position of the other letter but also repeated letters.
U have done 3*2 which includes all of them .
AT FIRST I COULDN'T UNDERSTAND then did this steps to understand

 

[JeffM]

@JeffM
Yes this is also a quick method .
I never thought of it.
I have the option of choosing either a or b.
1+1=2 options with any ONE OF LETTER which I will repeat twice .
So now I have any one position left out of 3
for the other letter.
For each of two positions occupied I have 1 place left.
Event 1:
If aa is at (1,3) then b is at 2nd place (1*1) that is I have 1 way of selecting the position of aa happening simultaneously with the event of selecting the position of other letter that why multiplication in between
So this was one whole Event/one unique arrangement, then right?
Or(+)
This is another event below which will happen if other events does not happen.

If aa is at (1,2) then b is at 3rd(1*1).
THIS IS another way of selecting the position of aa simultaneously happening with choosing the position of b
Or
This (below) is also one mutually exclusive event which will happen if no other events happen.
Thats why we use addition in between every events.

If aa is at (2,3) then b is at 1st(1*1)

So For (aa)we see that there are 3 exclusive events (1*1+1*1+1*1)
Similarly for (BB) there will be 3 exclusive events
Either one of 3 aa event or one of 3 BB event will happen =(1*1+1*1+1*1+1*1+1*1+1*1)

6 events all total
This is what you are trying to say by "How many positions can be chosen for the other letter"
I see that we are not only choosing the position of the other letter but also repeated letters.
U have done 3*2 which includes all of them .
AT FIRST I COULDN'T UNDERSTAND then did this steps to understand

Glad it helped.

Once you do enough of these problems, you will learn to see many ways to do them efficiently.