Sum of Product of Binomial Coefficients

Q.

Define a relation $\mathrm{R}$ over a class of $\mathrm{n}\times \mathrm{n}$ real matrices $\mathrm{A}\&\mathrm{B}$ as “$\mathrm{ARB}$ if there exists a non-singular matrix $\mathrm{P}$ such that $PA{P}^{-1}=B$”.

Then which of the following is true?

1. $\mathrm{R}$ is reflexive, symmetric but not transitive

2. $\mathrm{R}$ is symmetric, transitive but not reflexive

3. $\mathrm{R}$ is an equivalence relation

4. $\mathrm{R}$ is reflexive, Transitive but not symmetric

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. $20$

Q. The sum of $\sum _{r=0}^n(-1{)}^r\frac{{}^n{C}_r}{{}^{r+2}{C}_r}$ is

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

Q.

Find the sum of the series ${}^n{C}_0{}^n{C}_2+{}^n{C}_1{}^n{C}_3+{}^n{C}_2{}^n{C}_4.....{}^n{C}_{n-2}{}^n{C}_n$

1. ${}^{2n}{C}_{n-1}$

2. ${}^n{C}_{n-2}$

3. ${}^{2n}{C}_{n-2}$

4. $\frac{{}^{2n}{C}_{n-2}}{2}$

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. If ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\dots ,{}^n{C}_n$ denote the binomial coefficients in the expansion of $(1+x{)}^n$ and $p+q=1,$ then $\sum _{r=0}^n{r}^2{}^n{C}_r{p}^r{q}^{n-r}$ is

1. $np$
2. $npq$
3. $n^2{p}^2+npq$
4. None of these

Q. If ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\cdots ,{}^n{C}_n$ denote the binomial coeffients in the expansion of $(1+x{)}^n$ and $p+q=1,$ then $\sum _{r=0}^{n}r{}^n{C}_r{p}^r{q}^{n-r}$ is

1. $n{p}^2$
2. $npq$
3. $np$
4. None of these

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 ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\cdots ,{}^n{C}_n$ denote the binomial coeffients in the expansion of $(1+x{)}^n$ and $p+q=1,$ then $\sum _{r=0}^{n}r{}^n{C}_r{p}^r{q}^{n-r}$ is

1. $n{p}^2$
2. $npq$
3. $np$
4. None of these

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. Term independent of $x$ in the expansion of ${(x-\frac{1}{x^2})}^{3n}$ (where $n$ is even natural number)

1. $\frac{3n!}{n!\times 2n!}$
2. $\frac{3n!}{n!}$
3. $\frac{3n!}{2n!}$
4. $\frac{3n!}{(n!{)}^2}$

Q. For $r=0,1,...,10$, let $A_r,B_r,$ and $C_r$ denote, respctively, the coefficient of $x^r$ in the expansions of $(1+x{)}^{10},(1+x{)}^{20}$ and $(1+x{)}^{30}$. Then ${\sum }_{r=1}^{10}{A}_r(B_{10}{B}_r-C_{10}{A}_r)$is equal to

1. $B_{10}-C_{10}$
2. $A_{10}\left(B_{10}^2{C}_{10}{A}_{10}\right)$
3. \N
4. $C_{10}-B_{10}$

Q. The sum of the series
$2\cdot {}^{20}{C}_0+5\cdot {}^{20}{C}_1+8\cdot {}^{20}{C}_2+11\cdot {}^{20}{C}_3+\cdots +62\cdot {}^{20}{C}_{20}$
is equal to:

1. $2^{24}$
2. $2^{25}$
3. $2^{23}$
4. $2^{26}$

Q. The sum of the series
$2\cdot {}^{20}{C}_0+5\cdot {}^{20}{C}_1+8\cdot {}^{20}{C}_2+11\cdot {}^{20}{C}_3+\cdots +62\cdot {}^{20}{C}_{20}$
is equal to:

1. $2^{24}$
2. $2^{25}$
3. $2^{23}$
4. $2^{26}$

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. $na(n+1)$