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How to solve this factorial equation?

$$\frac{n!}{(n-6)!} = 350.418$$

or to give you the original equation:

$$0.146 = \binom{n}{6} \times 0.45^6 \times 0.55^{n-6}$$

Sorry, but I've no idea about LaTex

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  • $\begingroup$ Are we allowed to assume $n$ is an integer? $\endgroup$ – Gerry Myerson Mar 31 at 4:27
  • $\begingroup$ Perhaps you have in mind interpreting factorial for noninteger values of the argument using the gamma function. Alternatively you might think of $n!/(n-6)!$ as a descending product of six terms in $n$ as a polynomial. Perhaps you should explain what motivates "the original equation" so Readers can better advise you. $\endgroup$ – hardmath Mar 31 at 4:29
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    $\begingroup$ Looking at the second equation, it seems that it is using the Binomial distribution and that the first equation does not corresponds to the same thing. $\endgroup$ – Ertxiem Mar 31 at 4:29
  • $\begingroup$ Regarding LaTex, you may find some examples in this tutorial. $\endgroup$ – Ertxiem Mar 31 at 4:31
  • $\begingroup$ WolframAlpha does not offer an integer solution to this. If the $0.146$ is merely an approximation, that could be a cause. The closest solution to an integer is $n \approx 16.889 \approx 17$ so that might be it? $\endgroup$ – Eevee Trainer Mar 31 at 4:36
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Here's the binomial distribution under your parameters. The closest answer is $n=17$ where the distribution value is $0.143168$:

enter image description here

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