Question 11
Find the number of triples of nonnegative integers $(a, b, c)$ satisfying $a + b + c = 300$ and $$
\begin{aligned} &a^2 b + a^2 c \\ &\quad {}+ b^2 a + b^2 c \\ &\quad {}+ c^2 a + c^2 b = 6{,}000{,}000. \end{aligned}
$$
\begin{aligned} &a^2 b + a^2 c \\ &\quad {}+ b^2 a + b^2 c \\ &\quad {}+ c^2 a + c^2 b = 6{,}000{,}000. \end{aligned}
$$