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

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SUMMARY

The discussion focuses on simplifying the combination formula \binom{n}{\frac{n}{2}} using the factorial formula \frac{n!}{(\frac{n}{2})!(\frac{n}{2})!}. The user derived an expression involving 2^{n} and a series of terms (1-\frac{1}{n})(1-\frac{1}{n-2})(1-\frac{1}{n-4})... but sought further simplification. Participants recommended utilizing Stirling's approximation to analyze the behavior of the formula as n increases, emphasizing its importance in probability theory.

PREREQUISITES
  • Understanding of combination formulas and factorials
  • Familiarity with Stirling's approximation
  • Basic knowledge of probability theory
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Research Stirling's approximation and its applications in combinatorics
  • Explore the asymptotic behavior of \binom{n}{\frac{n}{2}} as n approaches infinity
  • Learn about the implications of large n in probability distributions
  • Study advanced combinatorial identities and their proofs
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Students of mathematics, particularly those studying combinatorics and probability, as well as educators looking to deepen their understanding of factorials and approximation techniques.

Aleoa
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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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