જો  $(\mathrm{x}+3)^{\mathrm{n}-1}+(\mathrm{x}+3)^{\mathrm{n}-2}(\mathrm{x}+2)+ $ $ (\mathrm{x}+3)^{\mathrm{n}-3}(\mathrm{x}+2)^2+\ldots . .+(\mathrm{x}+2)^{\mathrm{n}-1}$ માં $x^r$ નો સહગુણક $\alpha_{\mathrm{r}}$ છે. જો $\sum_{\mathrm{r}-0}^{\mathrm{n}} \alpha_{\mathrm{r}}=\beta^{\mathrm{n}}-\gamma^{\mathrm{n}}, \beta, \gamma \in \mathrm{N}$, તો $\beta^2+\gamma^2=$……………… 

Let n and k be positive integers such that $n \ge \frac{{k(k + 1)}}{2}$. The number of solutions $({x_1},{x_2},….{x_k})$, ${x_1} \ge 1,{x_2} \ge 2,….{x_k} \ge k,$ all integers, satisfying ${x_1} + {x_2} + …. + {x_k} = n$, is

$\sum_{ r =0}^{6}\left({ }^{6} C _{ r }{ }^{-6} C _{6- r }\right)$ ની કિમંત મેળવો.

જો $\left({ }^{30} C _1\right)^2+2\left({ }^{30} C _2\right)^2+3\left({ }^{30} C _3\right)^2+\ldots \ldots+30\left({ }^{30} C _{30}\right)^2=$ $\frac{\alpha 60 !}{(30 !)^2}$ હોય,તો $\alpha=…………$

$^n{C_1}\sum\limits_{r = 0}^1 {^1{C_r}} { + ^n}{C_2}\left( {\sum\limits_{r = 0}^2 {^2{C_r}} } \right){ + ^n}{C_3}\left( {\sum\limits_{r = 0}^3 {^3{C_r}} } \right) + ……{ + ^n}{C_n}\left( {\sum\limits_{r = 0}^n {^n{C_r}} } \right)$ ની કિમત મેળવો