For positive integers \(A\) and \(B\), \(1 \leq A \leq B \leq 100 \), suppose that \(A\) distinct integers are randomly chosen among the first \(B\) positive integers. Let \(C\) be the smallest integer among the \(A\) integers chosen. How many pairs \( ( A,B) \) are there such that the expected value of \(C\) is greater than or equal to \(10\)?

This problem is posed by Derek K.

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