Series involving Binomial Coefficients

$\frac{{}^{15}C_0}{6},\frac{{}^{15}C_1}{8},\frac{{}^{15}C_2}{10},....\frac{{}^{15}C_{15}}{38}$ is a bino-harmonic series.

Summation of bino-harmonic series $\frac{{}^{n}C_0}{a},\frac{{}^{n}C_1}{a+d},\frac{{}^{n}C_2}{a+2d},....\frac{{}^{n}C_n}{a+nd}$ is given by 

Express the summation of bino-harmonic series $\frac{{}^{15}C_0}{6},\frac{{}^{15}C_1}{8},\frac{{}^{15}C_2}{10},....\frac{{}^{15}C_{15}}{36}$ in the integral form.

Express the summation of bino-harmonic series $\frac{{}^{23}C_0}{4},\frac{{}^{23}C_1}{7},\frac{{}^{23}C_2}{10},....\frac{{}^{23}C_{23}}{73}$ in the integral form.

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

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 $\sum _{r=0}^{16}{a}_{3r}=$

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

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

Cotyledons are also called-

The sum $S=\frac{1}{9!}+\frac{1}{3!7!}+\frac{1}{5!5!}+\frac{1}{7!3!}+\frac{1}{9!}$ equals

If $C_r$ denotes the binomial coefficient ${}^n{C}_r$ then $\left(-1\right)C_0^2+2C_1^2+5C_2^2+\dots +\left(3n-1\right)C_n^2=$

The sum $\underset{0\le \text{i}<\text{j}\le 10}{\sum \sum }\left({}^{10}{\text{C}}_{\text{j}}\right)\left({}^{\text{j}}{\text{C}}_{\text{i}}\right)$ is equal to

If $N$ is a prime number which divides $S{=}^{39}{P}_{19}{+}^{38}{P}_{19}{+}^{37}{P}_{19}+.....{+}^{20}{P}_{19}$ , then the largest possible value of $N$ among following is

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+ \dots +C_n{x}^n$, then the value of $\sum _{0 \le r \le }\sum _{s \le n}\left(r+s\right) \left(C_r+C_s\right)$ is 

The value of $\frac{C_0}{1.3}-\frac{C_1}{2.3}+\frac{C_2}{3.3}-\frac{C_3}{4.3}+\dots +{\left(-1\right)}^n \frac{C_n}{\left(n+1\right).3}$ is

The A.M. of ${}^n{C}_0{,}^n{C}_1{,}^n{C}_2 ,\dots {,}^n{C}_n$ is: