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Can someone explain how this:

$$n!\cdot(n + 2)!$$

could become:

$ n\cdot(n + 2)!\cdot(n - 1)!$

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2 Answers 2

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$$ n! = n \cdot (n - 1)! $$ That's all.

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  • $\begingroup$ So easy but I couldn't see it. Thanks a lot :) $\endgroup$
    – DimChtz
    Commented Aug 25, 2016 at 19:50
  • $\begingroup$ This assumes n > 0 right? $\endgroup$
    – jlars62
    Commented Aug 25, 2016 at 21:03
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    $\begingroup$ @jlars62, The gamma function defined by $\Gamma(t) = \int_0^\infty x^{t-1}e^{-x}\,dx$ satisfies the functional equation $\Gamma(t+1) = t\Gamma(t)$, and is itself an extension of the factorial function for nonnegative integers. See here: en.wikipedia.org/wiki/Gamma_function. There are definitions of the gamma function that make use of negative integers as well. However, my answer speaks to $n \gt 0$, yes. $\endgroup$
    – Alex Ortiz
    Commented Aug 25, 2016 at 21:16
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You have

$$n!=1\times 2\times\cdots\times n$$

so

$$n!=(1\times\cdots \times n-1)\times n=(n-1)!\times n.$$

Therefore

$$n!\cdot (n+2)!=n\cdot (n-1)!\cdot (n+2)! = n\cdot (n+2)!\cdot (n-1)!.$$

And that's it.

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