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I know this is a silly and simple question, but I am confused by the symbol $\binom{n}{r}$. I am studying combinatorics. In the book, it uses $\binom{n}{r}$ in the following three situations: $$\binom{n}{r}=\frac{n!}{r!(n-r)!}---(combination)$$ $$\binom{n}{r}=\frac{n!}{(n-r)!}----(permutation)$$ $$\binom{n}{r}=\frac{n!}{r!}-----(?)$$

I think the correct way should be $$\binom{n}{r}=\frac{n!}{r!(n-r)!}---(combination)$$ $$P(n,r)=\frac{n!}{(n-r)!}---(permutation)$$

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    $\begingroup$ The common usage is the first one. I have never seen the other two. $\endgroup$ – Phira Mar 2 '17 at 14:12
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    $\begingroup$ I would say the second two are simply wrong. $\endgroup$ – lulu Mar 2 '17 at 14:14
  • $\begingroup$ What book are you using? $\endgroup$ – Sean English Mar 2 '17 at 14:22
  • $\begingroup$ Can you tell use which book and cite an example of where it uses the incorrect thing? $\endgroup$ – arctic tern Mar 2 '17 at 14:22
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Your first definition is also named $ C(n,r) = \frac{n!}{r!(n-r)!} = \begin{pmatrix}{n \\ r} \end{pmatrix}$ This is called a Combination and is the number of NON-ordered Committees of size $r$ you can make out of a set size $n$. Remember C = Combination = Committee.

The second definition is also named $P(n,r) = \frac{n!}{(n-r)!}$. The bracket form is NOT used for this. $P(n,r)$ is called a Permutation and it is the number of ordered Parades of length $r$ you can make out of a set of size $n$. Remember P = Permutation = Parade.

The formula $\frac{n!}{r!}$ is non standard and while it may come up in various work it doesn't have a specific name. The bracket form does NOT mean this.

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Only the first one.

There is a technical term for the other two: Wrong.

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