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This is a relatively elementary question, but the pdf I'm reading is confusing me.

In how many ways can we choose $3$ objects in a circle of $28$ objects? As a circle is invariant under rotation, if we choose to number the objects $1$ to $28$, I can always assume that I've chosen the object at position $1$. Then the other two objects can be selected in ${27\choose 2}=351$ ways.

However, the pdf that I'm consulting says that the number of ways should be ${28\choose 3}$. Where am I going wrong?

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    $\begingroup$ Not following. Of course there are $\binom {28}3$ ways to choose $3$ distinguishable objects from $28$. What has the circle got to do with anything? $\endgroup$ – lulu Mar 30 '18 at 13:56
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    $\begingroup$ You seem to be confusing arrangements, in which the relative order matters, with subsets, in which it does not. $\endgroup$ – N. F. Taussig Mar 30 '18 at 13:58
  • $\begingroup$ I always just thought that because a circle is invariant under rotations, we can always fix a position. For example, the number of ways of selecting $1$ object from a circle of $28$ would be $1$, as on selecting any other object we can just rotate the circle until we get the selected object to the $1$ position $\endgroup$ – fierydemon Mar 30 '18 at 14:00
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    $\begingroup$ Fixing a position doesn't mean choosing an object. The invariance of circle under rotation is significant when you are arranging things not when you are choosing. $\endgroup$ – Vidyanshu Mishra Mar 30 '18 at 14:05
  • $\begingroup$ @VidyanshuMishra- I find that strange. For instance, if the circle is invariant under rotation, then choosing the objects $\{4,5,6\}$ would be the same as choosing the objects $\{1,2,3\}$. The invariance should not be relevant only for arrangements $\endgroup$ – fierydemon Mar 30 '18 at 14:09
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To clarify the doubts you have raised in the question and in further comments,
the question should really have specified distinct objects, if they expect an answer of $\binom{28}3$


Added remarks in view of multifarious comments

  • If the objects are distinct, the answer is $\binom{28}3$ as given in the book.

  • If the objects are identical, your answer of $\binom{27}2$ is correct.

  • Since the question is not specific on this point, I'd say that the prudent thing to do is to either work out the answer under each assumption, or to point out the ambiguity in the question, and explicitly state the assumption under which you have worked out the answer

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Edit: I forgot the non-trivial case of $(0,\pi/n,2\pi/n)$

If I understand you well, you just want the symmetric property of a circle be taken to count, I answered a question like this before.

Take a random point from a circle. the ways of choosing 3 referred to a fixed point is $\binom{27}{2}$

There is a case of duplicity (or symmetry) of choices for both two points chosen in function of the distance between all three points, shown in red to be similar.

enter image description here

which means there is $\frac{\color{blue}{\binom{27}{2}}-\color{red}{3\frac{26}{2}}}{\color{blue}4}+\color{red}{\frac{26}{2}}=91$ ways. AFAIK and according to your subjective view apart from what the paper says.

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"As a circle is invariant under rotation, if we choose to number the objects 1 to 28, I can always assume that I've chosen the object at position 1. "

Imagine this: Mr. Left drove his car 5 miles. Then he drove it 3 miles. How many miles did Mr. Left drive.

As his name is Mr. Left the number of miles he drives will how many miles he has left after subtraction. And as $5 - 3 = 2$ he will have drive $2$ miles.

That logic makes as much sense as yours does. What does a circle be invariant under rotation have to do with anything?

Suppose that in position 1 was a bag of dog turds, and around on the other places were diamond rings and gold bullion, Rolex Watches, a first edition of Finnegan' Wake, and other useful things. You start by picking a diamond ring. But the guy who set this up says, "No, you have to choose the bag of dog turds first". You ask why and he says "Because the circle is invariant under rotation."

You give him dirty look, and pick the diamond ring in position 4, a micky mantle rookie card in position 5, and the topkapi daggar in position 6.

So the says "4,5,6. That's really the same thing as 1,2,3. So you picked the bag of dog turds, a bag of Channukkah chocolate, and a tuna sandwich made with truffle oil."

Eventually, it's time to walk away from this lunatic.

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  • $\begingroup$ I can't tell in what is the bloody utility of the arrangement shape taken by these objects,they can be circle or line or whatever since the objects are distinct, otherwise if they are not, shape has great significance. $\endgroup$ – Abr001am Mar 30 '18 at 17:07
  • $\begingroup$ Shape has the significance you give it. If the question else that all the objects were exactly the same and they were placed in a huge sack "how many way are there to choose three". I guess the answer to that is $1$; anyway you choose three is the same. It seems there is this babyfood pablum way of teaching that is to teach key words. "circle" means no distinction among rotation. But that is corner-cutting pablum. You don't assume that just because the problem mentioned "circle". You assume that if and only if it is indicated in the question. There is NOTHING is the question about varie $\endgroup$ – fleablood Mar 31 '18 at 0:50
  • $\begingroup$ you haven't understood me; i know my answer contradicts the basis where if all objects are distinct (and it's worth a vote-down), just i wonder what is the significance of circle shape since all objects are disticnt ? they can be aligned in a row or sacked in "pablum" or wtv, the answer still holds same = (3 choose 28). $\endgroup$ – Abr001am Mar 31 '18 at 1:50
  • $\begingroup$ Well, that's the thing. Unless the question specifies the circle is specific there is no significance to it, any more than say, a bag of objects being blue. My complaint is that this "rote-decypher-the-keyword:'circle'-means-rotationally-invariant" is just plain wrong. You don't determine how to do the problem based on it having the word "circle" in it, you determine how to do the problem based on what the problem actual is. Which could include a circle not being invariant, or a main named Mr. Left who doesnt do subtraction. $\endgroup$ – fleablood Mar 31 '18 at 5:38
  • $\begingroup$ The objects may be indistinct. So the only way to distinguish the objects is by position. So imagine that instead of bag of turds, and a bag of diamonds etc, you have $28$ bags of turds arranged in a circle. Also, the positions are not marked. In how many ways can you select three bags of turds? This is where the fact that the circle is invariant under rotation becomes relevant $\endgroup$ – fierydemon Mar 31 '18 at 5:50

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