Series involving Binomial Coefficients

Let $n>2$ be an integer and define a polynomial $p\left(x\right)=x^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$, where $a_0,a_1,\dots ,a_{n-1}$ are integers. Suppose we know that $np\left(x\right)=\left(1+x\right)p^{\text{'}}\left(x\right)$. If $b=p\left(1\right)$, then

$\left({}_{ }{}^7{C}_0+{}^7{C}_1\right)+\left({}^7{C}_2+{}^7{C}_3\right)+\dots +\left({}^7{C}_6+{}^7{C}_7\right)=$

The sum of the series ${2.}^{20}{C}_0+{5.}^{20}{C}_1+{8.}^{20}{C}_2+{11.}^{20}{C}_3+.......+{62.}^{20}{C}_{20}$ is equal to

The sum of the series ${2.}^{20}{C}_0+{5.}^{20}{C}_1+{8.}^{20}{C}_2+{11.}^{20}{C}_3+.......+{62.}^{20}{C}_{20}$ is equal to

Find the sum of the coefficient of all the integral power of $x$ in the expansion of ${\left(1+\frac{\sqrt{x}}{2}\right)}^{40}$

If $\text{'}n\text{'}$ is a positive integer, then $\sum _{r=1}^n{r}^2·C_r=\left(\_\_\_\_\_\right)2^{n-2}$

If $n\in N$ and $\left(1-2x+5x^2+10x^3\right)(1+x{)}^n=a_0+a_{1}x+a_2{x}^2+\dots$, and $a_1^2=2{\mathrm{a}}_2$, then value of $a_0$$+n$ is

If ${\left(1+x+x^2\right)}^{25}=\underset{r=0}{\overset{50}{\Sigma }}{a}_r{x}^r$, then

 $\underset{r=0}{\overset{12}{\Sigma }}{a}_{4r}=$

If ${\left(1+x+x^2\right)}^{25}=\underset{r=0}{\overset{50}{\Sigma }}{a}_r x^r$, then $\sum _{r=0}^{16}{a}_{3r}=$

If $\sum _{r=1}^9{\left(\frac{r+3}{2^r}\right)}^9{C}_r=\alpha {\left(\frac{3}{2}\right)}^9+\beta ,$ then $\alpha +\beta$ is equal to

Statement- $1$: $\sum _{r=0}^n(r+1{)}^n{C}_r=(n+2)2^{n-1}$
Statement -$2$:$\sum _{r=0}^n(r+1{)}^n{C}_r{x}^r=(1+x{)}^n+nx(1+x{)}^{n-1}$

The value of ${}^{50}C_4+\sum _{r=1}^6{}^{56-r}C_3$ is :
 

For any positive integer $m, n \left(with n\ge m\right),let\left(\begin{array}{l}n\\ m\end{array}\right)={}^{n}C_m.$ Prove that $\left(\begin{array}{c}n\\ m\end{array}\right)+\left(\begin{array}{c}n-1\\ m\end{array}\right)+\left(\begin{array}{c}n-2\\ m\end{array}\right)+\dots +\left(\begin{array}{c}m\\ m\end{array}\right)=\left(\begin{array}{c}n+1\\ m+1\end{array}\right)·$ Hence or otherwise, prove that $\left(\begin{array}{l}n\\ m\end{array}\right)+2\left(\begin{array}{c}n-1\\ m\end{array}\right)+3\left(\begin{array}{c}n-2\\ m\end{array}\right)+\dots +\left(n-m+1\right)\left(\begin{array}{c}m\\ m\end{array}\right)=\left(\begin{array}{c}n+2\\ m+2\end{array}\right)$.

For $2\le r\le n,\left(\begin{array}{l}n\\ r\end{array}\right)+2\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{c}n\\ r-2\end{array}\right)=$

The value of sum of the series $3{·}^n{C}_0+\frac{{\mathrm{ }}^n{C}_1\cdot 3^2}{2}+\frac{{\mathrm{ }}^n{C}_2\cdot 3^3}{3}+\frac{{\mathrm{ }}^n{C}_3\cdot 3^4}{4}+...\frac{{\mathrm{ }}^n{C}_n\cdot 3^{n+1}}{n+1}$ is

Evaluate :  ${{}^{20}C}_0-{{}^{20}C}_1+{{}^{20}C}_2-{{}^{20}C}_3+\text{...}-\text{...}+{{}^{20}C}_{10}$.

The value of ${\left({ }^n{C}_0\right)}^2+{\left({ }^n{C}_1\right)}^2+{\left({ }^n{C}_2\right)}^2+...+{\left({ }^n{C}_n\right)}^2=$_____

If ${\left(1+x+x^2\right)}^{25}=a_0+a_{1}x+a_2{x}^2+.......+a_{50}{x}^{50}$,  then $a_0+a_2+a_4+......+a_{50}$ is ______

Value of ${}^{10}\mathrm{C}_0{·}^{20}{\mathrm{C}}_{10}{+}^{10}{\mathrm{C}}_1{·}^{20}{\mathrm{C}}_9+\dots {+}^{10}{\mathrm{C}}_{10}{·}^{20}{\mathrm{C}}_0$ 

The value of ${}^{n}C_0^2+{}^{n}C_1^2+{}^{n}C_2^2+\dots +{}^{n}C_n^2$ is