# Yay for 2014! #4

Let $$N$$ represent the number of ordered pairs of positive integers $$(x_{1}, x_{2}, ..., x_{201})$$ such that $$\sum_{i = 1}^{201} x_{i} = 2014$$, and, for some $$k \in \{1, 2, ..., 201\}$$, $$x_{k} \geq 14$$. If $$N = \binom{10m}{m}$$, for some positive integer $$m$$, find the hundreds digit of $$m$$.

Notes:

$$\sum_{n = 1}^{k} x_{n} = x_{1} + x_{2} + ... + x_{k}$$.

$$\binom{n}{k} = _{n}C_{k}$$ (combination).

This problem is part of the set Yay for 2014!.

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