Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3, 7, 11, \ldots$, if $40 < \frac{6}{n(n+1)}\sum_{k=1}^{n} S_k < 42$, then $n$ equals $\underline{\hspace{1cm}}$.

Let $\alpha = \sum _ { r = 0 } ^ { n } \left( 4 r ^ { 2 } + 2 r + 1 \right) ^ { n } C _ { r }$ and $\beta = \left( \sum _ { r = 0 } ^ { n } \frac { { } ^ { n } C _ { r } } { r + 1 } \right) + \frac { 1 } { n + 1 }$. If $140 < \frac { 2 \alpha } { \beta } < 281$, then the value of $n$ is $\_\_\_\_$

Let $\alpha , \beta , \gamma$ and $\delta$ be the coefficients of $x ^ { 7 } , x ^ { 5 } , x ^ { 3 }$ and $x$ respectively in the expansion of $\left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , x > 1$. If u and v satisfy the equations $\begin{aligned} & \alpha u + \beta v = 18 \\ & \gamma u + \delta v = 20 \end{aligned}$ then $u + v$ equals :
(1) 5
(2) 3
(3) 4
(4) 8

Let the coefficients of three consecutive terms $T _ { r } , T _ { r + 1 }$ and $T _ { r + 2 }$ in the binomial expansion of $( a + b ) ^ { 12 }$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $( \sqrt [ 4 ] { 3 } + \sqrt [ 3 ] { 4 } ) ^ { 12 }$. Then $\mathrm { p } + \mathrm { q }$ is equal to :
(1) 283
(2) 287
(3) 295
(4) 299

For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $(1+x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+x)^{n+4}$ is:
(1) 20
(2) 10
(3) 35
(4) 70

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

The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7} + \sqrt[12]{11})^n$ is 183, is:
(1) 2184
(2) 2196
(3) 2148
(4) 2172

If in the expansion of $( 1 + x ) ^ { \mathrm { p } } ( 1 - x ) ^ { \mathrm { q } }$, the coefficients of $x$ and $x ^ { 2 }$ are 1 and $-2$, respectively, then $\mathrm { p } ^ { 2 } + \mathrm { q } ^ { 2 }$ is equal to :
(1) 18
(2) 13
(3) 8
(4) 20

If $\sum _ { r = 1 } ^ { 30 } \frac { r ^ { 2 } \left( { } ^ { 30 } C _ { r } \right) ^ { 2 } } { { } ^ { 30 } C _ { r - 1 } } = \alpha \times 2 ^ { 29 }$, then $\alpha$ is equal to $\_\_\_\_$

If $\sum _ { r = 0 } ^ { 5 } \frac { { } ^ { 11 } C _ { 2r } } { 2 r + 2 } = \frac { \mathrm { m } } { \mathrm { n } } , \operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } - \mathrm { n }$ is equal to $\_\_\_\_$

If $\alpha = 1 + \sum _ { r = 1 } ^ { 6 } ( - 3 ) ^ { r - 1 } \quad { } ^ { 12 } \mathrm { C } _ { 2 r - 1 }$, then the distance of the point $( 12 , \sqrt { 3 } )$ from the line $\alpha x - \sqrt { 3 } y + 1 = 0$ is $\_\_\_\_$.

The sum of all rational terms in the expansion of $\left( 1 + 2 ^ { 1 / 2 } + 3 ^ { 1 / 2 } \right) ^ { 6 }$ is equal to

Consider a polynomial in $x$ and $y$
$$P = ( 3 x + 4 y + 1 ) ^ { 5 } .$$
Let us denote the coefficient of $x ^ { n } y$ in the expansion of $P$ by $a _ { n }$, where $n$ is an integer. Note that $x ^ { 0 } = y ^ { 0 } = 1$.
(1) Let us find the value of the coefficient $a _ { 1 }$. First, we note that
$$P = \{ ( 3 x + 1 ) + 4 y \} ^ { 5 }$$
and use the binomial theorem to expand $P$. Then $x y$ appears when we expand the term AB $( 3 x + 1 ) ^ { \text {C} } y$. Further, the coefficient for $x$ in the expansion of $( 3 x + 1 ) ^ { \text {C} }$ is DE. It follows that
$$a _ { 1 } = \mathbf { F G H } .$$
(2) The number of values which $n$ can take is $\square$ in all. Also, the value of $a _ { n }$ is maximized at $n =$ $\square$ J .

$$P ( x ) = ( x + 2 ) ^ { 4 } + 3 ( x + 1 ) ^ { 3 }$$
In this polynomial, what is the coefficient of the $\mathbf { x }$ term?
A) 41
B) 39
C) 37
D) 35
E) 33

$$\mathrm { P } ( \mathrm { x } ) = ( \mathrm { x } - 1 ) ^ { 4 } + ( \mathrm { x } - 1 ) ^ { 5 }$$
In this polynomial, what is the coefficient of the $x ^ { 3 }$ term?
A) 4
B) 6
C) 9
D) 10
E) 11

For every subsets $A$ and $B$ of a non-empty set $X$, the operation $\odot$ is defined as
$$\mathrm { A } \odot \mathrm { B} = \mathrm { X } \backslash ( \mathrm { A} \cup \mathrm { B} )$$
For every subsets $K$ and $L$ of X satisfying the condition $K \subseteq L$, what is the result of the operation $$( \mathbf { X } \backslash \mathbf { L } ) \odot ( \mathbf { L } \backslash \mathbf { K } )$$
A) $X$
B) K
C) L
D) $X \backslash K$
E) $X \backslash L$

$$P ( x ) = ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) ^ { 4 }$$
What is the coefficient of the $x ^ { 4 }$ term in the polynomial?
A) 8
B) 10
C) 12
D) 14
E) 16

Where $m$ and $n$ are integers,
$$\left(x^2 + 2y\right)^7$$
In the expansion of this expression, if one of the terms is $mx^ny^2$, what is the sum $m + n$?
A) 56
B) 64
C) 72
D) 86
E) 94

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