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 $\sqrt { 30 }$

Find the coefficient of the $x ^ { 4 }$ term in the expansion of
$$x ^ { 2 } \left( 2 x + \frac { 1 } { x } \right) ^ { 6 }$$

Find the coefficient of $x$ in the expression:
$$(1+x)^0 + (1+x)^1 + (1+x)^2 + (1+x)^3 + \cdots + (1+x)^{79} + (1+x)^{80}$$

Find the coefficient of $x^2y^4$ in the expansion of $(1 + x + y^2)^7$
A $6$
B $10$
C $21$
D $35$
E $105$
F $210$

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

Find the coefficient of the $x ^ { 5 }$ term in the expansion of
$$( 1 + x ) ^ { 5 } \times \sum _ { i = 0 } ^ { 5 } x ^ { i }$$
A 1
B 5
C 16
D 25
E 32

In the simplified expansion of $( 2 + 3 x ) ^ { 12 }$, how many of the terms have a coefficient that is divisible by 12 ?

4 (See the solution/explanation page)

Let $n$, $k$ be integers satisfying $1 \leq k \leq n$. From the $n$ integers $2^m$ ($m = 0, 1, 2, \cdots, n-1$), choose $k$ distinct elements and take their product. Let $a_{n,k}$ denote the sum of the ${}_{n}C_{k}$ integers obtained by taking such products over all possible ways of choosing $k$ integers. For example,
$$a_{4,3} = 2^0 \cdot 2^1 \cdot 2^2 + 2^0 \cdot 2^1 \cdot 2^3 + 2^0 \cdot 2^2 \cdot 2^3 + 2^1 \cdot 2^2 \cdot 2^3 = 120$$

(1) For integers $n \geq 2$, find $a_{n,2}$.

(2) For integers $n \geq 1$, consider the polynomial in $x$: $$f_n(x) = 1 + a_{n,1}x + a_{n,2}x^2 + \cdots + a_{n,n}x^n$$ Express $\dfrac{f_{n+1}(x)}{f_n(x)}$ and $\dfrac{f_{n+1}(x)}{f_n(2x)}$ as polynomials in $x$.

(3) Express $\dfrac{a_{n+1,k+1}}{a_{n,k}}$ in terms of $n$ and $k$.
%% Page 5

$$( a + 1 ) ^ { 2 } - ( a - 1 ) ^ { 2 }$$
Which of the following is this expression equal to?
A) a
B) $2 a$
C) $3 a$
D) 4 a
E) $5 a$

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

$$\frac { [ ( n + 1 ) ! ] ^ { 2 } + ( n ! ) ^ { 2 } } { [ ( n + 1 ) ! ] ^ { 2 } - ( n ! ) ^ { 2 } } = \frac { 61 } { 60 }$$
Given this, what is n?
A) 9
B) 10
C) 12
D) 13
E) 15

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

$$P ( x ) = \left( x ^ { 2 } + 2 \right) ^ { 3 } + ( x - 3 ) ^ { 5 }$$
In this polynomial, what is the coefficient of the $x ^ { 4 }$ term?
A) - 9
B) - 3
C) 1
D) 11
E) 21

Let m and n be real numbers. In the expansion of
$$\left( \frac { \mathrm { m } } { \mathrm { nx } } + \frac { \mathrm { nx } ^ { 2 } } { \mathrm {~m} } \right) ^ { 3 }$$
arranged according to powers of x, the constant term is 6.
Accordingly, what is the ratio $\frac { m } { n }$?
A) 1
B) 2
C) 3
D) 4
E) 5

$$P ( x ) = ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) ^ { 4 }$$
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 $n$ 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