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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