If \(\sum _{r\text{ }=\text{ }1}^{9}\left(\frac{r+3}{{2}^{r}}\right)\text{ }.{\text{ }}^{9}{C}_{r}=\text{ }\alpha {\left…

If \(\sum _{r\text{ }=\text{ }1}^{9}\left(\frac{r+3}{{2}^{r}}\right)\text{ }.{\text{ }}^{9}{C}_{r}=\text{ }\alpha {\left(\frac{3}{2}\right)}^{9}−\text{ }\beta ,\alpha ,\beta \in \mathrm{N},\) then \((\alpha +\beta {)}^{2}\) is equal to

[JEE Main 2025, 3 Apr (Shift 1)]