MathNet 2010 — Part 02 Question 126

For any set $A = \{a_1, a_2, \dots, a_m\}$, denote $P(A) = a_1 a_2 \dots a_m$. Let $A_1, A_2, \dots$, and $A_n$ be all 99-element subsets of $\{1, 2, \dots, 2010\}$, $n = C_{2010}^{99}$. Prove that $2010 \mid \sum_{i=1}^{n} P(A_i)$.