For nonnegative integers a and b with a+b \leq 6, let T(a, b)=\left(\begin{array}{c}6 \\ a\end{array}\right)\left(\begin{array}{c}6 \\ b\end{array}\right)\left(\begin{array}{c}6 \\ a+b\end{array}\right). Let S denote the sum of all T(a, b), where a and b are nonnegative integers with a+b \leq 6. Find the remainder when S is divided by 1000 .