Solve the inequation: $\;$ $\dfrac{P^{n+1}_{4}}{C^{n-1}_{n-3}} > 14 \times P_3$, $\;$ $n \in N$
$\dfrac{P^{n+1}_{4}}{C^{n-1}_{n-3}} > 14 \times P_3$
i.e. $\;$ $\dfrac{\left(n + 1\right)!}{\left(n + 1 - 4\right)!} \div \dfrac{\left(n-1\right)!}{\left(n-3\right)! \left(n-1-n+3\right)!} > 14 \times 3!$
i.e. $\;$ $\dfrac{\left(n+1\right)!}{\left(n-3\right)!} \times \dfrac{\left(n-3\right)! \times 2!}{\left(n-1\right)!} > 14 \times 6$
i.e. $\;$ $\dfrac{2 \left(n+1\right) n \left(n-1\right)!}{\left(n-1\right)!} > 14 \times 6$
i.e. $\;$ $n \left(n+1\right) > 14 \times 3$
i.e. $\;$ $n^2 + n - 42 > 0$
i.e. $\;$ $\left(n+7\right) \left(n-6\right) > 0$
i.e. $\;$ $n+7 > 0$ $\;$ and $\;$ $n - 6 > 0$ $\;\;$ OR $\;\;$ $n + 7 < 0$ $\;$ and $\;$ $n - 6 < 0$
i.e. $\;$ $n > -7$ $\;$ and $\;$ $n > 6$ $\;\;$ OR $\;\;$ $n < -7$ $\;$ and $\;$ $n < 6$
i.e. $\;$ $n = \left\{-6, -5, -4, \cdots, 0, 1, 2, \cdots, 5, 6, 7, 8, \cdots\right\} \cap \left\{7, 8, 9, \cdots\right\}$
OR $\;\;$ $n = \left\{-8, -9, -10, \cdots\right\} \cap \left\{5, 4, 3, 2, 1, 0, -1, \cdots, -6, -7, -8, -9, \cdots\right\}$
i.e. $\;$ $n = \left\{7, 8, 9, \cdots\right\}$ $\;\;$ OR $\;\;$ $n = \left\{-8, -9, -10, \cdots\right\}$
$\because \;$ $n \in N$ $\implies$ $n = \left\{7, 8, 9, \cdots\right\}$