Sum of Product of Binomial Coefficients

Q. The value of ${\sum }_{r=0}^n(-1{)}^r\left(\frac{{}^n{C}_r}{{}^{r+3}{C}_r}\right)$ is

1. \N
2. $\frac{3!}{2(n+3)}$
3. $n$
4. $3!\left(n^2\right)$

Q.

If $\alpha$ not equal $1$ is any nth root of unity then $S=1+3\alpha +5{\alpha }^2+...$to $n$ terms equals

1. $\frac{2n}{(1-\alpha )}$

2. $-\frac{2n}{(1-\alpha )}$

3. $\frac{n}{(1-\alpha )}$

4. $-\frac{n}{(1-\alpha )}$

Q. The sum of the series $2C_0+\frac{C_1}{2}\cdot 2^2+\frac{C_2}{3}\cdot 2^3+\frac{C_3}{4}\cdot 2^4+\cdots +\frac{C_n}{n+1}\cdot 2^{n+1}$ is equal to , where$\left(C_r{=}^n{C}_r\right)$

1. $\frac{3^{n+1}-1}{n-1}$
2. $\frac{3^{n+1}-1}{n+1}$
3. $\frac{3^{n+1}+1}{n+1}$
4. $\frac{3^{n-1}-1}{n+1}$

Q. The value of the expression $C_0^2-C_1^2+C_2^2-\cdots +(-1{)}^n{C}_n^2$

1. 0 if n is odd
2. $(-1{)}^n$ if n is odd
3. $(-1{)}^{\frac{n}{2}}{}^n{C}_{\frac{n}{2}}$ if n is even
4. $(-1{)}^{n-1}{}^n{C}_{n-1}$ if n is even

Q.

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+........+C_n{x}^2$, then

$C_0^2+C_1^2+C_2^2+C_3^2+..........+C_n^2$ =

1. $\frac{n!}{n!n!}$

2. $\frac{\left(2n\right)!}{n!n!}$

3. $\frac{\left(2n\right)!}{n!}$

4. None of these

Q. If the sum of the series ${}^{12}{C}_5+7{.}^{12}{C}_6+21{.}^{12}{C}_7+35{.}^{12}{C}_8+35{.}^{12}{C}_9+21{.}^{12}{C}_{10}+{7.}^{12}{C}_{11}{+}^{12}{C}_{12}$ is equal to ${}^{2n+1}{C}_{12}$ then $n$ is

Q. The value of $\sum _{n=1}^6(3+2^n)$ is

1. $144$
2. $15$
3. $20$
4. $100$

Q. If $(1+x{)}^n=\sum _{r=0}^n{}^n{C}_r{x}^r$ and $\sum _{r=0}^n\frac{1}{{}^n{C}_r}=a,$ then the value of $\sum _{0\le i\le n}\sum _{0\le j\le n}(\frac{i}{{}^n{C}_i}+\frac{j}{{}^n{C}_j})$ is equal to

1. $n^{2}a$
2. $\frac{n^2}{2a}$
3. $\frac{na}{2}$
4. $na(n+1)$

Q. If $C_r$ stands for ${}^n{C}_r$ then the sum of the series
$\frac{2\left(\frac{n}{2}\right)!\left(\frac{n}{2}\right)}{n!}[C_0^2-2C_1^2+3C_2^2-\cdots +(-1{)}^n(n+1)C_n^2]$, where n is an even positive integer is equal to

1. \N
2. $(-1{)}^{\frac{n}{2}}(n+2)$
3. $(-1{)}^n(n+2)$
4. $(-1{)}^{n}n$

Q. The value of $r$ for which ${}^{20}{C}_r{}^{20}{C}_0+{}^{20}{C}_{r-1}{}^{20}{C}_1+{}^{20}{C}_{r-2}{}^{20}{C}_2+\dots +{}^{20}{C}_0{}^{20}{C}_r$ is maximum, is :

1. $11$
2. $15$
3. $10$
4. $20$