If Σ_r=1^30 r^2 ( ^30 C_r )^2 ^30 C_r-1=α × 2^29, then α is equal to

$\displaystyle \begin{aligned} & \sum_{\mathrm{r}=1}^{30} \frac{\mathrm{r}^2\left({ }^{30} \mathrm{C}_{\mathrm{r}}\right)^2}{{ }^{30} \mathrm{C}_{\mathrm{r}-1}} \\ & =\sum_{\mathrm{r}=1}^{30} \mathrm{r}^2\left(\frac{31-\mathrm{r}}{\mathrm{r}}\right) \cdot \frac{30!}{\mathrm{r}!(30-\mathrm{r})!} \\ & \left(\because \frac{{ }^{30} \mathrm{C}_{\mathrm{r}}}{{ }^{30} \mathrm{C}_{\mathrm{r}-1}}=\frac{30-\mathrm{r}+1}{\mathrm{r}}=\frac{31-\mathrm{r}}{\mathrm{r}}\right)\end{aligned}$​r=1∑30​30Cr−1​r2(30Cr​)2​=r=1∑30​r2(r31−r​)⋅r!(30−r)!30!​(∵30Cr−1​30Cr​​=r30−r+1​=r31−r​)​ $\displaystyle \begin{aligned} & =\sum_{\mathrm{r}=1}^{30} \frac{(31-\mathrm{r}) 30!}{(\mathrm{r}-1)!(30-\mathrm{r})!} \\ & =30 \sum_{\mathrm{r}=1}^{30} \frac{(31-\mathrm{r}) 29!}{(\mathrm{r}-1)!(30-\mathrm{r})!} \\ & =30 \sum_{\mathrm{r}=1}^{30}(30-\mathrm{r}+1)^{29} \mathrm{C}_{30-\mathrm{r}} \\ & =30\left(\sum_{\mathrm{r}=1}^{30}(31-\mathrm{r})^{29} \mathrm{C}_{30-\mathrm{r}}+\sum_{\mathrm{r}=1}^{30}{ }^{29} \mathrm{C}_{30-\mathrm{r}}\right) \\ & =30\left(29 \times 2^{28}+2^{29}\right)=30(29+2) 2^{28} \\ & =15 \times 31 \times 2^{29} \\ & =465\left(2^{29}\right) \\ & \alpha=465\end{aligned}$​=r=1∑30​(r−1)!(30−r)!(31−r)30!​=30r=1∑30​(r−1)!(30−r)!(31−r)29!​=30r=1∑30​(30−r+1)29C30−r​=30(r=1∑30​(31−r)29C30−r​+r=1∑30​29C30−r​)=30(29×228+229)=30(29+2)228=15×31×229=465(229)α=465​