Solve the inequation: $\;$ $C^{13}_{m} < C^{13}_{m + 2}$, $\;$ $m \in N$
$C^{13}_{m} < C^{13}_{m + 2}$
i.e. $\;$ $\dfrac{13!}{m! \left(13 - m\right)!} < \dfrac{13!}{\left(m + 2\right)! \left(13 - m - 2\right)!}$
i.e. $\;$ $\left(m + 2\right)! \left(11 - m\right)! < m! \left(13 - m\right)!$
i.e. $\;$ $\left(m + 2\right) \left(m + 1\right) m! \left(11 - m\right)! < m! \left(13 - m\right) \left(12 - m\right) \left(11 - m\right)!$
i.e. $\;$ $\left(m + 2\right) \left(m + 1\right) < \left(13 - m\right) \left(12 - m\right)$
i.e. $\;$ $m^2 + 3m + 2 < 156 - 25m + m^2$
i.e. $\;$ $28m < 154$
i.e. $\;$ $m < \dfrac{154}{28}$
i.e. $\;$ $m < \dfrac{11}{2}$
$\because \;$ $m \in N$ $\implies$ $m = \left\{1, 2, 3, 4, 5\right\}$