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! .