# Sum of the functions

Level pending

Disclaimer: I posted this problem earlier, but it had a wrong answer. This is the updated version of the problem.

Let $$x$$ and $$y$$ be integers such that $$1 < x \leq y$$.

Define $$f(x, y) = (1 - \dfrac{1}{x}) (1 - \dfrac{1}{x + 1}) (1 - \dfrac{1}{x + 2}) ... (1 - \dfrac{1}{y})$$

Let $$S = f(2, 2014) + f(3, 2014) + f(4, 2014) + ... + f(2013, 2014)$$

If $$S$$ can be expressed in the form $$\dfrac{a}{b}$$, where $$a, b$$ are coprime, positive integers, what is the last $$3$$ digits of $$a + b$$?

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