Series Involving Binomial Coefficients

Consider the following statements for a fixed natural number $n$ :

$1.$ $C\left(n, r\right)$ is greatest if $n=2r$

$2.$ $C\left(n, r\right)$ is greatest if $n=2r-1$ and $n=2r+1$

Which of the statements given above is/are correct?

The value of the expression $1^3.{{(}^n{C}_0)}^2+2^3.{{(}^n{C}_1)}^2+3^3{{(}^n{C}_2)}^2+......+n^3{{(}^n{C}_n)}^2$  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 $\sum _{r=1}^n{r}^2·C_r=\left(\_\_\_\_\_\right)2^{n-2}$

${}^{14}{\mathrm{C}}_7+\sum _{i=1}^3{ }^{17-\mathrm{i}}{\mathrm{C}}_6=$

Let ${\left(1-x-2x^2\right)}^6=1+a_{1}x+a_2{x}^2+\dots +a_{12}{x}^{12}$. Then $\frac{a_2}{2^2}+\frac{a_4}{2^4}+\frac{a_6}{2^6}+\dots$$\dots +\frac{a_{12}}{2^{12}}$ is equal to

If $(1+x{)}^n=\sum _{r=0}^n{}^{n}C_r{x}^r$, then

${}^{n}C_0+\frac{{}^{n}C_1}{2}+\dots \frac{{}^{n}C_n}{n+1}$ is

Let ${\left(1+x+x^2\right)}^3(1+2x)=a_0+a_{1}x+a_2{x}^2+\dots \dots +a_7{x}^7$ then $\sum _{k=0}^7\frac{a_k}{k+1}$ is equal to

If ${\left(1+x+x^2\right)}^{100}=a_0+a_{1}x+a_2{x}^2+\dots \dots +a_{200}{x}^{200}$ where $a_0,a_1,a_2,\dots \dots ,a_{200}$ are real constant and $x$ is a variable then $a_0+a_1+a$, equals

If $a=\left({}^{100}C_0+{}^{100}C_3+{}^{100}C_6+\dots ..+{}^{100}C_{99}\right), b=\left({}^{100}C_1+{}^{100}C_2+{}^{100}C_3+\dots ..\right)$ and
$c=\left({}^{100}C_1-{}^{100}C_2+{}^{100}C_4-{}^{100}C_5+\dots ..\right)$, then the value of ${\left(a-\frac{b}{2}\right)}^2+\frac{3}{4}{c}^2$ is equal to

The value of $\frac{{\mathrm{C}}_1}{{\mathrm{C}}_0}+2·\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1}+3·\frac{{\mathrm{C}}_3}{{\mathrm{C}}_2}+\dots +\mathrm{n}\frac{{\mathrm{C}}_{\mathrm{n}}}{{\mathrm{C}}_{\mathrm{n}-1}}$ is equal to :

The value of ${}^{\mathit{n}}C_0+{}^{\mathit{n}}C_1+5{}^{\mathit{n}}C_2+11{}^{\mathit{n}}C_3+17{}^{\mathit{n}}C_4\dots \dots \dots .(n+1)$ terms

If $C_r$ stands for ${}^{n}C_r,$ then the coefficient of ${\lambda }^n{\mu }^n$ in the expansion of $\left[\left(1+\lambda \right)\left(1+\mu \right)\left(\lambda +\mu \right)\right]{}^{\mathrm{n}}$ is:

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

Let ${\left(1+x\right)}^{10}=\sum _{r=0}^{10}{C}_r{x}^r$ and ${\left(1+x\right)}^7=\sum _{r=0}^7{d}_r{x}^r$ If $P=\sum _{r=0}^5{C}_{2r}$ and $Q=\sum _{r=0}^3{d}_{2r+1},$ then $\frac{P}{2Q}$ is equal to-

If ${}^{17}\mathrm{C}_1{2}^{16}+{}^{17}\mathrm{C}_2{2}^{16}+3.{}^{17}\mathrm{C}_3{2}^{14}+4.{}^{17}\mathrm{C}_4·2^{13}+......+17.{}^{17}\mathrm{C}_{17}=\mathrm{x}.3^{\mathrm{y}}$ then $\mathrm{x}+\mathrm{y}$ is equal to

Let ${\left(1+x+x^2\right)}^{25}=a_0+a_{1}x+a_2{x}^2+\dots \dots +a_{50}{x}^{50}.$ If $A=\left(a_3-a_5+a_7-a_9+\dots \dots -a_{49}\right),$ then find $\frac{A}{8}$

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