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Is there a name or short way of writing of $\frac{n!}{m!}$? I've searched and the closest I could find was binomial coefficient. Is there any other way?

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That tends to be called a falling or rising factorial, and there are multiple notations, though none of them are standard enough that you could use one in a paper without defining it first.

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One way that I've seen is $(n)_m$. This is a common way to write this notion in combinatorics.

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    $\begingroup$ I would just like to point out that this is the notation for many operations, including the base number (a subscript 10 for base 10) and rows and columns in linear algebra (a subscript of 3,4 for row 3, column 4). $\endgroup$ – Joao May 13 '14 at 2:21

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