Coefficients of Terms Equidistant from Beginning and End

Q.

Number of ways of selection of 8 letters from 24 letters of which 8 are a, 8 are b and the rest unlike, is given by

1. 

2. 

3. 

4. 

Q. The given expression is ${(x+\sqrt{x^3-1})}^5+{(x-\sqrt{x^3-1})}^5$ is a

1. polynomial of fractional degree.
2. polynomial of degree 7.
3. polynomial with integer degree.
4. not a polynomial.

Q.

The coefficient of $x^4$ of the expression ( 1 + 5x +$9x^2+......\infty \left)\right(1+x^2{)}^{11}$ is

1. 

2. 

3. 

4. 

Q.

If the coefficient of $x^{7}in{(a{x}^2-\frac{1}{bx})}^{11}$ is equal to the coefficient of $x^{-7}in{(ax-\frac{1}{b{x}^2})}^{11}$ then ab =

1. 0

2. 1

3. -1

4. 2

Q. Expand the expression $(2x-3{)}^6$

Q.

Find the coefficient of $x^{90}$ in $(x+x^2+x^3+..........{)}^2(x^5+x^6+...........{)}^9$

1. 

2. 

3. 

4. 

Q. The coefficient of $x^{2}in(1+x)(1+2x)(1+4x)....(1+2^{n}x)$ is

1. 
2. 
3. 
4. 

Q.

The coefficient of $x^{10}$ in $(1+x+x^2+x^3{)}^5$ is

1. 98

2. 101

3. 100

4. 75

Q. If the ratio of the $5^{\text{th}}$ term from the beginning to the $5^{\text{th}}$ term from the end in the expansion of ${(\sqrt[4]{42}+\frac{1}{\sqrt[4]{43}})}^n$ is $\sqrt{6}:1,$ then the value of $n$ is

1. $12$
2. $10$
3. $8$
4. $14$

Q. In the expansion of $(1+a{)}^{m+n}$ prove that coefficients of $a^m$ and $a^n$ are equal.

Q.

One or more options can be correct:
Find the coefficient of $x^{10}$ in $(1-x^7)(1-x^8)(1-x^9)(1-x{)}^{-3}$

1. 

2. 

3. 

4. 

Q. The coefficient of $x^6$ in $\left\{\right(1+x{)}^6+(1+x{)}^7+.....+(1+x{)}^{1}5\}$is

1. 
2. 
3. 
4. 

Q.

The coefficient of $x^3$ in $(2x-\frac{3}{x^2}{)}^9$ is

1. 

2. 

3. 

4. 

Q.

Number of ways of selection of 8 umbrella from 24 umbrellas of which 8 are green color, 8 are blue color and the rest unlike, is given by

1. 

2. 

3. 

4. 

Q.

If the coefficient of $x^7$ and $x^8$ in ${(2+\frac{x}{3})}^n$ are equal, then n is:

1. 56

2. 55

3. 45

4. 15

Q. The coefficients of three consecutive terms of $(1+x{)}^{n+5}$ are in the ratio 5 : 10 : 14. Then, n is equal to

1. 5
2. 6
3. 3
4. 4

Q. The coefficient of $x^{3}in{(\sqrt{x^5}+\frac{3}{\sqrt{x^3}})}^{6}is$

1. 120
2. 420
3. 540
4. 0

Q. The value of \(\sum^{50}_{r=0}(^{100}C_r.^{200}C_{150+r})\) is equal to

1. 
2. 
3. 
4. 

Q. Coefficient of $t^{24}in(1+t^2{)}^{12}(1+t^{12}\left)\right(1+t^{24})$ is ?

1. ${}^{12}{C}_6+3$
2. ${}^{12}{C}_6+1$
3. ${}^{12}{C}_6$
4. ${}^{12}{C}_6+2$

Q. If the coefficients of $x^3$ and $x^4$ in the expansion of $(1+ax+b{x}^2$) $(1–2x{)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

1. $(16,\frac{251}{3})$
2. $(14,\frac{251}{3})$
3. $(14,\frac{272}{3})$
4. $(16,\frac{272}{3})$

Q.

Find the term independent of x in the expansion of $(2x^2+\frac{1}{x}{)}^9$

1. 

2. 

3. 

4. 

Q. The value of ${(}^{21}{C}_1{-}^{10}{C}_1)+{(}^{21}{C}_2{-}^{10}{C}_2)+{(}^{21}{C}_3{-}^{10}{C}_3)+{(}^{21}{C}_4{-}^{10}{C}_4)+...+{(}^{21}{C}_{10}{-}^{10}{C}_{10})$ is

1. $2^{21}-2^{11}$
2. $2^{21}-2^{10}$
3. $2^{20}-2^9$
4. $2^{20}-2^{10}$

Q.

One or more options can be correct.

Find the coefficient of $x^{100}$ in $(1-x{)}^{-3}$.

1. 

2. 

3. 

4. 

Q.

The number of ways in which an examiner can assign 30 marks to 8 questions, awarding not less than 2 marks to any question is ________.

1. 

2. 

3. None of these

4. 

Q. The ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of ${(2^{\frac{1}{3}}+\frac{1}{2(2{)}^{\frac{1}{3}}})}^{10}$ is:

1. $1:2(6{)}^{\frac{1}{3}}$
2. $2(36{)}^{\frac{1}{3}}:1$
3. $4(36{)}^{\frac{1}{3}}:1$
4. $1:4(16{)}^{\frac{1}{3}}$

Q. Find the coefficient of $a^5{b}^7$ in $(a-2b{)}^{12}$

1. 12356
2. 101376
3. -101356
4. -101376