Sum of Binomial Coefficients with Alternate Signs

Q. Find $r$ if $\left(i\right){}^5{P}_r=2{}^6{P}_{r-1}\left(ii\right){}^5{P}_r={}^6{P}_{r-1}$

Q. Find the value of $n$ such that
$\left(i\right){}^n{P}_5=42{}^n{P}_3,n>4$
$\left(ii\right)\frac{{}^n{P}_4}{{}^{n-1}{P}_4}=\frac{5}{3},n>4$

Q. Find $n$ if ${}^{n-1}{P}_3:{}^n{P}_4=1:9.$

Q. The coefficient of $x^k$ in $1+(1+x)+(1+x{)}^2+\cdots +(1+x{)}^n$, where $0\le k\le n$ is

1. ${}^{n-1}{C}_{k-1}$
2. ${}^n{C}_k$
3. ${}^{n+1}{C}_{k+1}$
4. ${}^{n+1}{C}_k$

Q. If ${(1+x+x^2)}^8=a_0+a_{1}x+a_2{x}^2+\cdots +a_{16}{x}^{16}$ for all real $x,$ then $a_5$ is equal to

Q. If ${}^n{P}_r={}^n{P}_{r+1}$ and ${}^n{C}_r={}^n{C}_{r-1}$, then the value of $r$ is equal to

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

Q.

If $f\left(x\right)={(a-x^n)}^{1/n}$, where $a>0$ and $n\in N$, then $fof\left(x\right)$ is equal to:

1. $a$

2. $x$

3. $x^n$

4. $a^n$

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. 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. Let $a_1,a_2,a_3\dots ,a_n$ be in A.P. and $a_3,a_5,a_8,b_1,b_2,b_3,\dots ,bn$ be in G.P. If $a_9=40,$ then

1. $\sum _{i=1}^9{a}_i^2=6144$
2. $\sum _{i=1}^{\infty }\frac{1}{b_i}=\frac{1}{18}$
3. $\sum _{i=1}^9{a}_i^2=6278$
4. $\sum _{i=1}^{\infty }\frac{1}{b_i}=\frac{1}{28}$

Q. The value of ${}^{15}{C}_8{+}^{15}{C}_9{-}^{15}{C}_6{-}^{15}{C}_7$ is

Q. If $C_0,C_1,C_2,\dots ,C_n$ denotos the binomial coefficients in the expansion of $(1+x{)}^n,$ and $1^3\cdot C_1+2^3\cdot C_2+3^3\cdot C_3+\dots +n^3{C}_n=\frac{1}{\lambda }\left(n^2\right)(n+\mu )\cdot 2^n,$ then $\lambda +\mu =$

Q. The value of ${}^{15}{C}_0^2-{}^{15}{C}_1^2+{}^{15}{C}_2^2-\cdots -{}^{15}{C}_{15}^2$ is

1. $15$
2. $-15$
3. $0$
4. $51$

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

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

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. The value of $(\sqrt{2}+1{)}^6+(\sqrt{2}-1{)}^6$ is

Q. If $C_0,C_1,C_2,\cdots ,C_n$ are the binomial coefficients and $S=2\times C_1+2^3\times C_3+2^5\times C_5+\cdots$, then which of the following is/are correct?

1. When $n=50$, then $S=\frac{3^{50}-1}{2}$
2. When $n=50$, then $S=\frac{3^{50}+1}{2}$
3. When $n=101$, then $S=\frac{3^{101}+1}{2}$
4. When $n=101$, then $S=\frac{3^{101}-1}{2}$

Q. The number of positive terms in the sequence $\left\{X_n\right\},$
if $X_n=\frac{90}{7{(}^n{P}_n)}-\frac{{}^{n+4}{P}_3}{{}^{n+2}{P}_{n+2}},n\in N$ is

1. $6$
2. $5$
3. $7$
4. $4$

Q. If $C_r$ denotes coefficient of $x^r$ in $(1+x{)}^{99}$ then the value of $C_0-2C_1+3C_2-4C_3+\dots 100C_{99}=$

Q. If ${}^{2n}{C}_2{+}^{2n}{C}_4{+}^{2n}{C}_6+\cdots +{}^{2n}{C}_{2n}=511$, then absolue value of ${}^n{C}_0-{}^n{C}_1{3}^1+{}^n{C}_2{3}^2+\cdots +(-1{)}^n{}^n{c}_n{3}^n$ is

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

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

Q. Evaluate ${\log }_{3}2{\log }_{4}3{\log }_5\mathrm{4.....}{\log }_{15}14{\log }_{16}15$

Q. If ${}^{2n}{C}_2{+}^{2n}{C}_4{+}^{2n}{C}_6+\cdots +{}^{2n}{C}_{2n}=511$, then absolue value of ${}^n{C}_0-{}^n{C}_1{3}^1+{}^n{C}_2{3}^2+\cdots +(-1{)}^n{}^n{c}_n{3}^n$ is

Q. For $n>0$, the value of ${}^n{C}_0-2^2\cdot {}^n{C}_1+3^2\cdot {}^n{C}_2-4^2{}^n{C}_3\cdots$ upto $(n+1)$ terms is $kn(n+1)\cdot 2^n,$ then $k=$

Q. $\frac{16\times 2^{n+1}-4\times 2^n}{16\times 2^{n+2}-2\times 2^{n+2}}$ equals

1. $\frac{1}{4}$
2. $-\frac{1}{2}$
3. $-\frac{1}{4}$
4. $\frac{1}{2}$

Q. If $x^2+\frac{1}{x^2}=23,$ evaluate $x+\frac{1}{x}$

Q. The value of ${}^{15}{C}_0^2-{}^{15}{C}_1^2+{}^{15}{C}_2^2-\cdots -{}^{15}{C}_{15}^2$ is

1. $15$
2. $-15$
3. $0$
4. $51$

Q. Value of $C_0+2C_1+3C_2+4C_3+\dots +(n+1)C_n$ is
( where $C_r={}^n{C}_r$)

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