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In the formula,

$\frac{n(n-1)(n-2)\cdots(n-r+2)}{(r-1)!}a^{n-r+1}b^{r-1}$

what does the "$\cdots$" mean?

3 Answers 3

13

It means: "There are too many terms to write, but follow the obvious pattern to fill them in".

In your example, you subtract $1$ from a factor to get the next factor. I might read that aloud as "$n$ times $n-1$ times $n-2$ all the way down to $n-r+2$".

As another example, $ 3 + 6 + 9 + \cdots + 3n $ would indicate the sum of all positive multiples of $3$ less than or equal to $3n$.

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    @Dan Not only may there be too many terms, but, the number of terms may not be fixed, e.g. $\,r-1\,$ terms in your product2012-09-04
2

Writing it as "$\cdots\;$", would be better than "$\dots\;$" . It indicates a product:

$ \frac{n(n-1)(n-2) \cdots (n-r+2)}{(r-1)!}a^{n-r+1}b^{r-1} =\frac{\prod_{k=0}^{r-2} (n-k)}{(r-1)!}a^{n-r+1}b^{r-1} $

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    Ah yeah. No problem.2012-09-04
0

It means multiplying a set of terms ($n$, $n-1$, $n-2$, $\dots$(!), $n-(r-2)$) when each term is the result of subtracting $0$, $1$, $2$, $\dots$, $r-2$ from $n$, respectively.