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How could we put a product of $n$ values, $x_{1},\, x_{2},\,\,\,\,\,....x_{n}$ into binomial coefficients. I mean $\prod\limits_{i=1}^{n}x_{i}=$ Some expression involving binomial coefficients. Note that all $x_{i}$ are non-negative real numbers and having the relation $x_{1}\geq x_{2}\geq x_{3}....\geq x_{n}$. Relation to Gamma function or factorial function will also be helpful. Thanks

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  • $\begingroup$ it is not clear what you mean by "into binomial coefficients" $\endgroup$ – Masacroso May 23 '17 at 9:27
  • $\begingroup$ How about $$\prod_{i=1}^{n}{\binom{x_i}{1}}?$$ $\endgroup$ – Jose Arnaldo Bebita-Dris May 23 '17 at 9:27
  • $\begingroup$ Yes, it works but now actually i want to relate this to factorial function or Gamma function in which i am facing trouble, any idea, Dear @Jose Arnaldo Bebita Dris. Thank you very much $\endgroup$ – Tahir Ullah Khan May 23 '17 at 14:12
  • $\begingroup$ If $n=1$ then the product is just $x_1$ which is equal to itself. What more do you want? $\endgroup$ – Somos May 23 '17 at 15:20
  • $\begingroup$ i want to replace this product by some expression involving Factorial function or Gamma function. As i have seen that this has a relation to factorial or gamma function but i dont know the actual way how to do this.@Somos $\endgroup$ – Tahir Ullah Khan May 23 '17 at 18:07

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