If $\sum _{\mathrm{k}=1}^{31} \left({ }^{31}{\mathrm{C}}_{\mathrm{k}}\right)\left({ }^{31}{\mathrm{C}}_{\mathrm{k}-1}\right)-\sum _{\mathrm{k}=1}^{30} \left({ }^{30}{\mathrm{C}}_{\mathrm{k}}\right)\left({ }^{30}{\mathrm{C}}_{\mathrm{k}-1}\right)=\frac{\mathrm{\alpha }(60!)}{(30!)(31!)}$, where $\mathrm{\alpha }\in \mathrm{ℝ}$, then the value of $16\mathrm{\alpha }$ is equal to

$\begin{array}{l}\sum _{\mathrm{k}=1}^{31}{ }^{31}{\mathrm{C}}_{\mathrm{k}}{\cdot }^{31}{\mathrm{C}}_{\mathrm{k}-1}{=}^{31}{\mathrm{C}}_1{\cdot }^{31}{\mathrm{C}}_0{+}^{31}{\mathrm{C}}_2{\cdot }^{31}{\mathrm{C}}_1+\dots .{+}^{31}{\mathrm{C}}_{31}{\cdot }^{31}{\mathrm{C}}_{30}\\ \\ {=}^{31}{\mathrm{C}}_0{\cdot }^{31}{\mathrm{C}}_{30}{+}^{31}{\mathrm{C}}_1{\cdot }^{31}{\mathrm{C}}_{29}+\dots {+}^{31}{\mathrm{C}}_{30}{\cdot }^{31}{\mathrm{C}}_0{=}^{62}{\mathrm{C}}_{30}\\ \end{array}$

$\begin{array}{l}\sum _{k=1}^{30} \left({ }^{30}{\mathrm{C}}_{\mathrm{k}}{\cdot }^{30}{\mathrm{C}}_{\mathrm{k}-1}\right){=}^{60}{\mathrm{C}}_{29}\\ \sum _{\mathrm{k}=1}^{31} \left({ }^{31}{\mathrm{C}}_{\mathrm{k}}\right)\left({ }^{31}{\mathrm{C}}_{\mathrm{k}-1}\right)-\sum _{\mathrm{k}=1}^{30} \left({ }^{30}{\mathrm{C}}_{\mathrm{k}}\right)\left({ }^{30}{\mathrm{C}}_{\mathrm{k}-1}\right){=}^{62}{\mathrm{C}}_{30}{-}^{60}{\mathrm{C}}_{29}=\frac{62!}{30!32!}-\frac{60!}{29!31!}\\ \Rightarrow \frac{\mathrm{\alpha }(60!)}{(30!)(31!)}=\frac{60!}{29!31!}\left\{\frac{62\cdot 61}{30\cdot 32}-1\right\}\\ \Rightarrow \frac{\mathrm{\alpha }(60!)}{(30!)(31!)}=\frac{60!}{30!31!}\left(\frac{2822}{32}\right)\\ \therefore 16\alpha =16\times \frac{2822}{32}=1411\end{array}$