Sum of Binomial Coefficients with Alternate Signs

Q. If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+C_3{x}^3+\cdots +C_n{x}^n,$ then $C_0{C}_2+C_1{C}_3+C_2{C}_4+\cdots +C_{n-2}{C}_n=$

1. $\frac{\left(2n\right)!}{(n!{)}^2}$
2. $\frac{\left(2n\right)!}{(n-1)!(n+1)!}$
3. $\frac{\left(2n\right)!}{(n-2)!(n+2)!}$
4. None of these

Q.

$C_0-C_1+C_2-C_3+........+(-1{)}^n{C}_n$ is equal to

1. $2^n$

2. $2^n-1$

3. 0

4. $2^{n-1}$

Q. The sum of co-efficients of all even degree terms in $x$ in the expansion of $(x+\sqrt{x^3-1}{)}^6+(x-\sqrt{x^3-1}{)}^6,(x>1)$ is equal to:

1. $24$
2. $26$
3. $29$
4. $32$

Q.

Let $S_1$ = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$.............${}^n{C}_n$ and $S_2$ = ${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$ ..............+ $(-1{)}^n$ ${}^n{C}_n$

Find the value of $\frac{S_1}{S_1+S_2}$ is ___.

Q. Sum of the series $3C_1-4C_2+5C_3-6C_4+\cdots$ upto $n$ terms is (where $C_r={}^n{C}_r$)

1. $-1$
2. $2$
3. $-2$
4. $1$

Q. Value of $C_0+2C_1+3C_2+4C_3+\dots +(n+1)C_n$ is
<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> ( where $C_r={}^n{C}_r$)

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

Q. If $C_0,C_1,C_2,\dots ,C_n$ denote the binomial coefficients in the expansion of $(1+x{)}^n,$ then value of $1^2\cdot C_1+2^2\cdot C_2+3^2\cdot C_3+\dots +n^2{C}_n$ is

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

Q. If $C_r$ represents ${}^{100}{C}_r,$ then $5C_0+8C_1+11C_2+\dots$ upto $101$ terms is equal to

1. $\left(305\right)\cdot 2^{100}$
2. $\left(305\right)\cdot 2^{99}$
3. $\left(310\right)\cdot 2^{100}$
4. $\left(310\right)\cdot 2^{99}$

Q.

The sum of the series ${}^{20}{C}_0$ - ${}^{20}{C}_1$ + ${}^{20}{C}_2$ - ${}^{20}{C}_3$ ...............+ ${}^{20}{C}_{10}$ is

1. ${}^{-20}{\text{C}}_{10}$

2. $\frac{1}{2}$${}^{20}{C}_{10}$

3. 0

4. ${}^{20}{C}_{10}$