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Putnam 1984/A5) Let

R

be the region consisting of all triples

(x,y,z)

of nonnegative real numbers satisfying

x+y+z\leq 1

. Let

w=1-x-y-z

. Express the value of the triple integral

\iiint_{R}xy^{9}z^{8}w^{4}\ dx\ dy\ dz

in the form

a!b!c!d!/n!

where

a,b,c,d

and

n

are positive integers. [hide="A solution"]\iiint_{R}xy^{9}z^{8}w^{4}\ dx dy dz = 4\iiint_{R}\int_{0}^{1-x-y-z}xy^{9}z^{8}w^{3}\ dw dx dy dz = 4\iiiint_{Q}xy^{9}z^{8}w^{3}\ dw dx dy dz
where

Q=\left\{ (x,y,z,w)\in\mathbb{R}^{4}|\ x,y,z,w\geq 0, x+y+z+w\leq 1\right\}

, which is a Dirichlet integral giving
4\iiiint_{Q}x^{1}y^{9}z^{8}w^{3}\ dw dx dy dz = 4\cdot\frac{1!9!8!3!}{(2+10+9+4)!}= \frac{1!9!8!4!}{25!}

Problem Statement