## Problem of the Day #421: Dim Sum Night
*December 30, 2012*

*Posted by Saketh in : potd , trackback*

Anand is eating at a Chinese restaurant with his family. He is going to order for all of them; he will submit one ordered tuple of non-negative integers $(x,y,z)$ representing the number of dumplings, spring rolls, and steamed buns he wants.

Now, well distributed orders (like $(10, 10, 10)$) are typically more satisfactory than unbalanced orders (such as $(29, 1, 0)$). After all, we don’t want everyone fighting over the lone spring roll! Let the quality $Q$ of a given order be modeled by the function $Q(x,y,z) = xyz$.

Anand knows how much his family will eat, so he wants $x+y+z = 30$. He will select a tuple $(x,y,z)$ at random from all those that satisfy this constraint. Determine the expected value of $Q(x,y,z)$, the quality of the order.

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