Discussion
Dirichlet's integral \( \displaystyle\iiint x^{l-1}y^{m-1}z^{n-1}\,dV \) over the simplex \( x+y+z \le 1 \) with all variables positive equals
Answer: C
The result is
\[ \frac{\Gamma(l)\Gamma(m)\Gamma(n)}{\Gamma(l+m+n+1)}, \]
and the extra unit in the denominator's argument is what distinguishes it from the beta-function pattern. Setting \( l = m = n = 1 \) gives \( 1/\Gamma(4) = 1/6 \), the volume of the simplex, which is the check that fixes the form
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