Mock AIME 2 2010 Problems/Problem 12

Problem

Let $K = \sum_{a_1=0}^8\sum_{a_2=0}^{a_1}...\sum_{a_{100}=0}^{a_{99}}\binom{8}{a_1}\binom{a_1}{a_2}...\binom{a_{99}}{a_{100}}.$ Find the sum of digits of $K$ in base-100.

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Mock AIME 2 2010 Problems/Problem 12

Problem

Solution