Mathematica returns values for non-integer arguments passed to Binomial.
What is the definition of Binomial for such continuous arguments?
Example:
Binomial[9, 2]
36
Binomial[9, 2.3212312]
49.801
Binomial[9, 3]
84
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Mathematica defines Binomial for non-integer inputs as follows:
$$ \binom{n}{m} = \frac{\Gamma(n+1)}{\Gamma(m+1)\Gamma(n-m+1)} $$
You'll find this under the Details section on the documentation page of Binomial.
Binomialfor non-integer arguments? Or are you asking how this value is computed? The definition is in the documentation under details. I don't know how it's computed, but it's probably not directly using the formula in terms of $\Gamma$ functions as that would result in the ratio of very large numbers. $\endgroup$ – Szabolcs May 17 '13 at 14:22LogGamma[]are used before exponentiating. $\endgroup$ – J. M.'s technical difficulties♦ May 17 '13 at 15:28