Sum of product of binomial coefficients

Q.

Write the following cube in expanded form: ${\left(2a-3b\right)}^3$

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.

$6^{th}$ term in expansion of $(2x^2-\frac{1}{3x^2}{)}^{10}$ is

1. 

2. 

3. 

4. None of these

Q.

The Derivative of $e^x$

1. $e^x$

2. $e^{x+1}$

3. $x{e}^x$

4. $e^{2x}$

Q.

If $x^4$ occurs in the $r^{th}$ term in the expansion of $(x^4+\frac{1}{x^3}{)}^{15}$, then r =

1. 7

2. 8

3. 9

4. 10

Q.

In the expansion of $(x+a{)}^n$, if the sum of odd terms be P and sum of even terms be Q. Consider the following statements:

(i) $P^2-Q^2=(x^2-a^2{)}^n$

(ii) $P^2+Q^2=(x^2+a^2{)}^n$

(iii) 4PQ = $(x+a{)}^{2n}-(x-a{)}^{2n}$

(iv) P - Q = $(x-a{)}^n$

Then the correct statement are:

1. (i), (iii), (iv)

2. (i), (iii)

3. (ii), (iii)

4. (i), (iv)

Q. ${(}^{10}{C}_0{)}^2-{(}^{10}{C}_1{)}^2+....-{(}^{10}{C}_9{)}^2+{(}^{10}{C}_{10}{)}^{2}equals$

1. 
2. 
3. 
4. 

Q.

The fourth term in the expansion of $(1-2x{)}^{\frac{3}{2}}$ will be

1. 

2. 

3. 

4. 

Q.

If ${}^n{C}_8={}^n{C}_6,$ find ${}^n{C}_3.$

Q. The 10th term in the expansion of $(x–1{)}^{11}$ (in decreasing powers of x) is

1. 
2. 
3. 
4. 

Q.

If a, b, and c are in A.P., p, q, and r are in H.P. and ap, bq, and cr are in G.P., then $\frac{p}{r}+\frac{r}{p}$ is equal to

1. 

2. 

3. 

4. 

Q.

${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$ - ${}^n{C}_3$.............. =

1. 0

2. 1

3. 

4. 

Q. For any positive integer $m,n$ with $n\ge m,$ let $\left(\begin{array}{c}n\\ m\end{array}\right)={}^n{C}_m.$ Then
$\left(\begin{array}{c}n\\ m\end{array}\right)+\left(\begin{array}{c}n-1\\ m\end{array}\right)+\left(\begin{array}{c}n-2\\ m\end{array}\right)+\cdots +\left(\begin{array}{c}m\\ m\end{array}\right)$ is equal to

1. ${}^n{C}_m$
2. ${}^{n+1}{C}_m$
3. ${}^n{C}_{m+1}$
4. ${}^{n+1}{C}_{m+1}$

Q.

If $T_r$ denotes the rth term in the expansion of ${(x+\frac{1}{y})}^{23}$ then

1. 

2. 

3. 

4. 

Q.

Find the value of ${\sum }_{i=1}^{\infty }$ ${\sum }_{j=1}^{\infty }$ ${\sum }_{k=1}^{\infty }$ $\frac{1}{2^i{2}^j{2}^k}$.

1. 4

2. 27

3. 8

4. 16

Q.

Which of the following is/are true?

(1) $(28-1{)}^{867}=7p-1$, where p is a positive integer

(2) $(63+1{)}^{563}=7q+1$, where q is a positive integer

1. 1&2

2. only 2

3. only 1

4. None of them

Q.

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

1. 

2. 

3. 

4. 0

Q. If the sum of odd terms and the sum of even terms in the expansion of ${(x+a)}^n$ are p and q respectively then
$p^2+q^2$

1. 
2. 
3. 
4. 

Q.

The value of $\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}$ is

1. 0

2. 

3. 

4. 

Q.

$r^{th}$ term in the expansion of $(a+2x{)}^n$ is

1. None of these

2. 

3. 

4. 

Q.

If the third term in the expansion of ${(\frac{1}{x}+x^{log}{10}^x)}^5$ is 1000, then x =

1. 

2. 

3. 

4. 

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. 0
3. $C_{10}-B_{10}$
4. $A_{10}\left(B_{10}^2{C}_{10}{A}_{10}\right)$

Q. Find the coefficient of $x^5$ in $(x+3{)}^8$

Q. Find $a$ if the coefficients of $x^2$ and $x^3$ in the expansion of $(3+ax{)}^9$ are equal.