I know that the highest power of a prime which divides $n!$ is given by
$$\left[\frac np\right]+\left[\frac n{p^2}\right]+\left[\frac n{p^3}\right]...$$
Where $[x]$ is the greatest integer function.
For $^{n}P_{r}$, which equals $\frac{n!}{(n-r)!}$, I can find the highest power of $p$ dividing $n!$ and $(n-r)!$, and then subtract the two to get the highest power of $p$ dividing $^{n}P_{r}$.
Now is there a better method to do this, where we don't need to find the highest power of $p$ dividing $(n-r)!$? Can we directly find the power of $p$ dividing the product of $r$ consecutive integers?
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