Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1 + x)^{2\mathrm{n}-1}$. If $2\mathrm{A} = 5\mathrm{B}$, then n is equal to:
(1) 22
(2) 20
(3) 21
(4) 19

Q64. If the coefficients of $x ^ { 4 } , x ^ { 5 }$ and $x ^ { 6 }$ in the expansion of $( 1 + x ) ^ { n }$ are in the arithmetic progression, then the maximum value of $n$ is:
(1) 7
(2) 21
(3) 28
(4) 14

Q83. If the second, third and fourth terms in the expansion of $( x + y ) ^ { n }$ are 135,30 and $\frac { 10 } { 3 }$, respectively, then $6 \left( n ^ { 3 } + x ^ { 2 } + y \right)$ is equal to $\_\_\_\_$

Q64. If the term independent of $x$ in the expansion of $\left( \sqrt { \mathrm { a } } x ^ { 2 } + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 10 }$ is 105 , then $\mathrm { a } ^ { 2 }$ is equal to :
(1) 2
(2) 4
(3) 6
(4) 9

In the expansion of $\left( \left( 1 + x ^ { 2 } \right) ^ { 2 } ( 1 + x ) ^ { n } \right.$, coefficients of $x , x ^ { 2 }$ and $x ^ { 3 }$ are in A.P, then find sum of all possible values of $n \in N$.

It is given that the expansion of $( a x + b ) ^ { 3 }$ is $8 x ^ { 3 } - p x ^ { 2 } + 18 x - 3 \sqrt { 3 }$, where $a , b$ and $p$ are real constants.
What is the value of $p$ ?
A $- 12 \sqrt { 3 }$ B $- 6 \sqrt { 3 }$ C $- 4 \sqrt { 3 }$ D $- \sqrt { 3 }$ E $\sqrt { 3 }$ F $4 \sqrt { 3 }$ G $6 \sqrt { 3 }$ H $12 \sqrt { 3 }$

The coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + 2 x + 3 x ^ { 2 } \right) ^ { 6 }$ is equal to twice the coefficient of $x ^ { 4 }$ in the expansion of $\left( 1 - a x ^ { 2 } \right) ^ { 5 }$.
Find all possible values of the constant $a$.
A $\pm 2 \sqrt { 2 }$ B $\pm \sqrt { 17 }$ C $\pm \sqrt { 34 }$ D $\pm 2 \sqrt { 17 }$ E There are no possible values of $a$.

In the expansion of $( a + b x ) ^ { 5 }$ the coefficient of $x ^ { 4 }$ is 8 times the coefficient of $x ^ { 2 }$.
Given that $a$ and $b$ are non-zero positive integers, what is the smallest possible value of $a + b$ ?
A 3
B 4
C 5
D 9
E 13
F 17

The non-zero constant $k$ is chosen so that the coefficients of $x ^ { 6 }$ in the expansions of $\left( 1 + k x ^ { 2 } \right) ^ { 7 }$ and $( k + x ) ^ { 10 }$ are equal.
What is the value of $k$ ?
A $\frac { 1 } { 6 }$
B 6
C $\frac { \sqrt { 6 } } { 6 }$
D $\sqrt { 6 }$
E $\frac { \sqrt { 30 } } { 30 }$
F $\sqrt { 30 }$

Consider the expansion of
$$( a + b x ) ^ { n }$$
The third term, in ascending powers of $x$, is $105 x ^ { 2 }$ The fourth term, in ascending powers of $x$, is $210 x ^ { 3 }$ The fourth term, in descending powers of $x$, is $210 x ^ { 3 }$ Find the value of $\frac { a } { b } ^ { 2 }$
A $\frac { 1 } { 4 }$
B $\frac { 4 } { 9 }$
C $\frac { 25 } { 36 }$
D $\frac { 5 } { 6 }$
E 1

Let $n$ be a natural number. Given that the arithmetic mean of all coefficients in the expansion of
$$\left( x ^ { 3 } - \frac { 2 } { x ^ { 2 } } \right) ^ { n }$$
is 0.2, what is the coefficient of the $x ^ { 2 }$ term in this expansion?
A) 12
B) 16
C) 24
D) 32
E) 40

Let $a$ be a positive real number,
$$\left(x + \frac{a - 7}{x}\right)^{13}$$
In the expansion of this expression, the coefficient of the $x^{11}$ term is $\frac{234}{a}$.
Accordingly, what is $a$?
A) 9 B) 12 C) 13 D) 15 E) 18

Let $m$ and $n$ be natural numbers. If the constant term in the expansion of
$$\left(x + \frac{5}{x^{m}}\right)^{n}$$
is 60, what is $m + n$?
A) 36 B) 35 C) 31 D) 27 E) 23

Let $n$ be a positive integer. In the expansion of
$$\left(x^{2} + x\right)^{n}$$
both the coefficient of the term containing $x^{19-n}$ and the coefficient of the term containing $x^{16-n}$ equal a positive integer $k$. Accordingly, what is $k$?
A) 6 B) 12 C) 15 D) 18 E) 21