# How to write "n choose k" as x^n y^k z^(n-k) without using Stirling

• October 4th 2012, 06:23 AM
gerritgroot
How to write "n choose k" as x^n y^k z^(n-k) without using Stirling
Hi,

Is there a way to write the combinatorial (n choose k) as:

${n \choose k} =Const. \cdot x^n y^k z^{n-k}$

Without using Stirling's formula and without having a product or summation in x, y and z?

What could x=x(n,k), y=y(n,k) and z=z(n,k) be?

Thanks,

Gerrit
• October 6th 2012, 06:03 AM
chiro
Re: How to write "n choose k" as x^n y^k z^(n-k) without using Stirling
Hey gerritgroot.

You may be able to use an integral representation, but the integral itself won't have a nice analytic answer if you try and evaluate it. If you are still interested, you probably want to check out the Beta Integral:

Beta function - Wikipedia, the free encyclopedia
• October 6th 2012, 09:41 AM
gerritgroot
Re: How to write "n choose k" as x^n y^k z^(n-k) without using Stirling
Thanks, I will check that