Best fit curve associated with the combination formula

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SUMMARY

The discussion centers on deriving a continuous expression for the discrete combination formula (denoted as ##_nC_r##). It is established that the best fit curve for this formula is a parabolic shape, which scales with the variable ##n##. The conversation highlights the importance of the gamma function as a generalization of factorials to real numbers, which is crucial for transitioning from discrete to continuous forms.

PREREQUISITES
  • Understanding of combinatorial mathematics, specifically the combination formula ##_nC_r##.
  • Familiarity with parabolic curves and their properties in mathematical analysis.
  • Knowledge of the gamma function and its role in extending factorials to real numbers.
  • Basic calculus concepts related to continuous functions and curve fitting.
NEXT STEPS
  • Research the properties and applications of the gamma function in combinatorial contexts.
  • Explore methods for curve fitting, particularly for parabolic functions.
  • Study the relationship between discrete and continuous probability distributions.
  • Investigate advanced topics in mathematical analysis related to best fit curves.
USEFUL FOR

Mathematicians, statisticians, and students studying combinatorial mathematics or curve fitting techniques will benefit from this discussion.

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Sorry if this is more of a HW question (if so then moderator please move my question. Thanks!)

Hi, I'm trying to get an expression for a best fit curve of the combination formula (##_nC_r##).
As far as I can tell, the curve is a simple parabolic curve, and its shape doesn't change. It's just scaled depending on ##n##.

How might I get the continuous form of the discrete combination formula?
 
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It is not a parabola.

The generalization of factorials to the real numbers is the gamma function.
 
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