The number of ways of choosing $3$ objects from $28$ in a circle.

Question

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?

Answer 1

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 $\binom{28}3$

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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

Answer 2

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.

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.

Answer 3

"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.