Simplifying Combination Formula: \binom{n}{\frac{n}{2}} using Factorial Formula

[Aleoa]

I'm trying to simplify the combination defined as : $\binom{n}{\frac{n}{2}}$.

I did some calculations, starting from the factorial formula $\frac{n!}{(\frac{n}{2})!(\frac{n}{2})!}$ and i found this form :

$2^{n}(1-\frac{1}{n})(1-\frac{1}{n-2})(1-\frac{1}{n-4})...$

but i don't know how to continue, can you help me ?

 

[fresh_42]

but i don't know how to continue, can you help me ?

No, since you haven't said what your goal is. To me $\binom{n}{\frac{n}{2}}$ is already fine.

 

[Aleoa]

I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support

 

[fresh_42]

I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support

In this case I'd try where Stirling's approximation would get me.

 

[Ray Vickson]

I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support

As fresh_42 suggested, use Stirling's formula. Every student of probability should be thoroughly familiar with that formula, as it is used everywhere.