Sum of Binomial Coefficients with Alternate Signs

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. If $C_0,C_1,C_2,\dots ,C_n$ denote the binomial coefficients respectively in $(1+x{)}^{2020},$ then

1. $C_0-\frac{C_1}{2}+\frac{C_2}{3}-\frac{C_3}{4}+\dots +\frac{C_{2020}}{2021}=\frac{1}{2021}$
2. $C_0+\frac{C_1}{2}+\frac{C_2}{3}+\frac{C_3}{4}+\dots +\frac{C_{2020}}{2021}=\frac{2^{2020}-1}{2021}$
3. $C_0+\frac{C_2}{3}+\frac{C_4}{5}+\dots +\frac{C_{2020}}{2021}=\frac{2^{2020}}{2021}$
4. $\frac{C_1}{2}+\frac{C_3}{4}+\dots +\frac{C_{1999}}{2020}=\frac{2^{2020}}{2021}$

Q. The value of ${}^{30}{C}_0-\frac{{}^{30}{C}_1}{2}+\frac{{}^{30}{C}_2}{3}-\cdots \cdots +\frac{{}^{30}{C}_{30}}{31}$ is

1. $\frac{1}{30}$
2. $\frac{30}{31}$
3. $\frac{31}{30}$
4. $\frac{1}{31}$

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. The value of $\sum _{r=0}^{50}(-1{)}^r\frac{{}^{50}{C}_r}{r+2}$ is equal to

1. $\frac{1}{50\times 51}$
2. $\frac{1}{52\times 50}$
3. $\frac{1}{52\times 51}$
4. $\frac{1}{50\times 53}$

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.

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