$\begin{aligned} \alpha= & \sum_{k=0}^n \frac{{ }^n C_k \cdot{ }^n C_k}{k+1} \cdot \frac{n+1}{n+1} \\ & =\frac{1}{n+1} \sum_{k=0}^n{ }^{n+1} C_{k+1} \cdot{ }^n C_{n-k} \\ \alpha & =\frac{1}{n+1} \cdot{ }^{2 n+1} C_{n+1} \\ \beta & =\sum_{k=0}^{n-1} C_k \cdot \frac{{ }^n C_{k+1}}{k+2} \frac{n+1}{n+1} \\ & \frac{1}{n+1} \sum_{k=0}^{n-1}{ }^n C_{n-k} \cdot{ }^{n+1} C_{k+2} \\ & =\frac{1}{n+1} \cdot{ }^{2 n+1} C_{n+2} \\ \frac{\beta}{\alpha} & =\frac{2 n+1}{2 n+1} C_{n+2} \\ \frac{\beta}{\alpha} & =\frac{2 n+1-(n+2)+1}{n+2}=\frac{5}{6} \\ n & =10 \end{aligned}$