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I understand that a combination is a way of selecting a finite number of things out of a larger group, in which the order of elements do not matter. That is all fine. But, in my Mathematical Reasoning course, we have come across using non-integers in the Binomial Theorem.

For example, in determining a finite number of terms of the expansion $(1+x)^{-\frac{1}{3}}$ one has to use the equation:

$$(1+x)^n=\sum_{i=0}^{\infty}{n\choose i}x^i$$

This clearly gives what is usually considered as a combination which one can not evaluate. But, avoiding the factorial definition of a combination allows us to assign a number to these coefficients.

My question is whether these combinations have any actually meaning in the traditionally combinatorial sense. Or, are they merely a means of expansion?

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Think about it - $\begin{pmatrix}n\\ k\end{pmatrix}$ means the number of ways of picking $k$ unordered outcomes from $n$ possibilities. –  Sanath Devalapurkar Feb 4 at 1:20
    
Yes, as I said in OQ, I am well aware of that fact. I'm trying to go deeper than that and find out what it means when they (n,k) are not integers. –  Richard P Feb 4 at 1:23
    
for your $x\choose i$ should it be $n\choose i$ –  Jay Feb 4 at 1:25
    
Yes, it should. Thanks for the catch. –  Richard P Feb 4 at 1:29
    
Would this be of any help ? –  Lucian Feb 4 at 2:27

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