# How to solve this factorial equation?

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

• Are we allowed to assume $n$ is an integer? – Gerry Myerson Mar 31 at 4:27
• 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. – hardmath Mar 31 at 4:29
• 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. – Ertxiem Mar 31 at 4:29
• Regarding LaTex, you may find some examples in this tutorial. – Ertxiem Mar 31 at 4:31
• 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? – Eevee Trainer Mar 31 at 4:36

Here's the binomial distribution under your parameters. The closest answer is $$n=17$$ where the distribution value is $$0.143168$$: 