# Super Duper Hedral Sum

**Discrete Mathematics**Level 5

\[ \large{\begin{eqnarray} a_{1,n} &=& 1+2+3+\ldots+n \\ a_{2,n} &=& a_{1,1} + a_{1,2} + a_{1,3} + \ldots + a_{1,n} \\ a_{3,n} &=& a_{2,1} + a_{2,2} + a_{2,3} + \ldots + a_{2,n} \\ a_{4,n} &=& a_{3,1} + a_{3,2} + a_{3,3} + \ldots + a_{3,n} \\ & \cdot & \\ & \cdot & \\ & \cdot & \\ a_{1000,n} &=& a_{999,1} + a_{999,2} + a_{999,3} + \ldots + a_{999,n} \\ \end{eqnarray}} \]

If we are given the 1000 equations above, evaluate the expression below.

\[ \large \left (1000 \prod_{k=1}^{1000} \frac{k+1000}k \right) \div a_{1000,1000} \]

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