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

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Homework Help Overview

The discussion revolves around simplifying the combination formula \(\binom{n}{\frac{n}{2}}\) using the factorial representation. Participants are exploring the implications of this simplification, particularly as \(n\) increases.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to simplify the combination formula starting from its factorial definition. There are questions about the next steps in the simplification process and the behavior of the formula as \(n\) grows larger. Some participants suggest using Stirling's approximation as a potential approach.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on simplification techniques and the relevance of Stirling's approximation. There is no explicit consensus on the best method to proceed, but there is a productive exchange of ideas regarding the characterization of the formula's behavior.

Contextual Notes

Some participants express concern about the clarity of their posts and the format of their questions, indicating a desire to improve communication in future discussions.

Aleoa
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Member has been warned not to delete the template.
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 ?
 
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Aleoa said:
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.
 
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
 
Aleoa said:
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.
 
Aleoa said:
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.
 

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