O valor de $\binom{10}{3}$ é
(A) 60 (B) 90 (C) 120 (D) 150 (E) 180

In a school project, João was invited to calculate the areas of several different squares, arranged in sequence, from left to right, as shown in the figure.
The first square in the sequence has a side measuring 1 cm, the second square has a side measuring 2 cm, the third square has a side measuring 3 cm, and so on. The objective of the project is to identify by how much the area of each square in the sequence exceeds the area of the previous square. The area of the square that occupies position $n$ in the sequence was represented by $\mathrm{A}_{n}$.
For $n \geq 2$, the value of the difference $\mathrm{A}_{n} - \mathrm{A}_{n-1}$, in square centimeter, is equal to
(A) $2n - 1$
(B) $2n + 1$
(C) $-2n + 1$
(D) $(n-1)^{2}$
(E) $n^{2} - 1$

Show that the power of $x$ with the largest coefficient in the polynomial $\left( 1 + \frac { 2 x } { 3 } \right) ^ { 20 }$ is 8 , i.e., if we write the given polynomial as $\sum _ { i } a _ { i } x ^ { i }$ then the largest coefficient $a _ { i }$ is $a _ { 8 }$.

In the expansion of the polynomial $2 ( x + a ) ^ { n }$, the coefficient of $x ^ { n - 1 }$ and the coefficient of $x ^ { n - 1 }$ in the expansion of the polynomial $( x - 1 ) ( x + a ) ^ { n }$ are equal. Find the maximum value of $a n$ for all ordered pairs $( a , n )$ satisfying this condition. (Here, $a$ is a natural number and $n$ is a natural number with $n \geqq 2$.) [4 points]

In the expansion of the polynomial $( x - a ) ^ { 5 }$, when the sum of the coefficient of $x$ and the constant term is 0, what is the value of the positive constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the coefficient of $x$ in the expansion of $\left( 2 x + \frac { 1 } { 2 x } \right) ^ { 7 }$? [3 points]
(1) 14
(2) 28
(3) 42
(4) 56
(5) 70

In the expansion of $\left( x + \frac { 1 } { x ^ { 3 } } \right) ^ { 4 }$, what is the coefficient of $\frac { 1 } { x ^ { 4 } }$? [4 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

The following is a proof by mathematical induction that the equality $$\sum _ { k = 0 } ^ { n } \frac { { } _ { n } \mathrm { C } _ { k } } { { } _ { n + 4 } \mathrm { C } _ { k } } = \frac { n + 5 } { 5 }$$ holds for all natural numbers $n$.
(1) When $n = 1$, $$( \text { LHS } ) = \frac { { } _ { 1 } \mathrm { C } _ { 0 } } { { } _ { 5 } \mathrm { C } _ { 0 } } + \frac { { } _ { 1 } \mathrm { C } _ { 1 } } { { } _ { 5 } \mathrm { C } _ { 1 } } = \frac { 6 } { 5 } , ( \text { RHS } ) = \frac { 1 + 5 } { 5 } = \frac { 6 } { 5 }$$ so the given equality holds.
(2) Assume that when $n = m$, the equality $$\sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } } = \frac { m + 5 } { 5 }$$ holds. When $n = m + 1$, $$\sum _ { k = 0 } ^ { m + 1 } \frac { { } _ { m + 1 } \mathrm { C } _ { k } } { { } _ { m + 5 } \mathrm { C } _ { k } } = \text { (가) } + \sum _ { k = 0 } ^ { m } \frac { { } _ { m + 1 } \mathrm { C } _ { k + 1 } } { { } _ { m + 5 } \mathrm { C } _ { k + 1 } }$$ For a natural number $l$, $${ } _ { l + 1 } \mathrm { C } _ { k + 1 } = \text { (나) } \cdot { } _ { l } \mathrm { C } _ { k } \quad ( 0 \leqq k \leqq l )$$ so $$\sum _ { k = 0 } ^ { m } \frac { { } _ { m + 1 } \mathrm { C } _ { k + 1 } } { { } _ { m + 5 } \mathrm { C } _ { k + 1 } } = \text { (다) } \cdot \sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } }$$ Therefore, $$\begin{aligned} \sum _ { k = 0 } ^ { m + 1 } \frac { { } _ { m + 1 } \mathrm { C } _ { k } } { { } _ { m + 5 } \mathrm { C } _ { k } } & = \text { (가) } + \text { (다) } \cdot \sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } } \\ & = \frac { m + 6 } { 5 } \end{aligned}$$ Thus, the given equality holds for all natural numbers $n$.
What are the correct values for (가), (나), and (다) in the above process? [3 points] $\begin{array} { l l l l } & \text { (가) } & \text { (나) } & \text { (다) } \\ \text { (1) } & 1 & \frac { l + 2 } { k + 2 } & \frac { m + 4 } { m + 4 } \end{array}$
(2) $1 \quad \frac { l + 1 } { k + 1 } \quad \frac { m + 1 } { m + 5 }$
(3) $1 \quad \frac { l + 1 } { k + 1 } \quad \frac { m + 1 } { m + 4 }$
(4) $m + 1 \quad \frac { l + 1 } { k + 1 } \quad \frac { m + 1 } { m + 5 }$
(5) $m + 1 \quad \frac { l + 2 } { k + 2 } \quad \frac { m + 1 } { m + 4 }$

The following is a proof by mathematical induction that the equality $$\sum _ { k = 0 } ^ { n } \frac { { } _ { n } \mathrm { C } _ { k } } { { } _ { n + 4 } \mathrm { C } _ { k } } = \frac { n + 5 } { 5 }$$ holds for all natural numbers $n$.

(1) When $n = 1$, $$( \text { Left side } ) = \frac { { } _ { 1 } \mathrm { C } _ { 0 } } { { } _ { 5 } \mathrm { C } _ { 0 } } + \frac { { } _ { 1 } \mathrm { C } _ { 1 } } { { } _ { 5 } \mathrm { C } _ { 1 } } = \frac { 6 } { 5 } , ( \text { Right side } ) = \frac { 1 + 5 } { 5 } = \frac { 6 } { 5 }$$ so the given equality holds.
(2) Assume that when $n = m$, the equality $$\sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } } = \frac { m + 5 } { 5 }$$ holds. When $n = m + 1$, $$\sum _ { k = 0 } ^ { m + 1 } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k } } = \text { (가) } + \sum _ { k = 0 } ^ { m } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k + 1 } }$$ For a natural number $l$, $${ } _ { l + 1 } \mathrm { C } _ { k + 1 } = \text { (나) } \cdot { } _ { l } \mathrm { C } _ { k } \quad ( 0 \leqq k \leqq l )$$ so $$\sum _ { k = 0 } ^ { m } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k + 1 } } = \text { (다) } \cdot \sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } }$$ Therefore, $$\begin{aligned} \sum _ { k = 0 } ^ { m + 1 } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k } } & = \text { (가) } + \text { (다) } \cdot \sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } } \\ & = \frac { m + 6 } { 5 } \end{aligned}$$ Thus, the given equality holds for all natural numbers $n$.
Which of the following are correct for (가), (나), and (다)? [3 points]

(가)	(나)	(다)
(1) 1	$\frac { l + 2 } { k + 2 }$	$\frac { m + 1 } { m + 4 }$
(2) 1	$\frac { l + 1 } { k + 1 }$	$\frac { m + 1 } { m + 5 }$
(3) 1	$\frac { l + 1 } { k + 1 }$	$\frac { m + 1 } { m + 4 }$
(4) $m + 1$	$\frac { l + 1 } { k + 1 }$	$\frac { m + 1 } { m + 5 }$
(5) $m + 1$	$\frac { l + 2 } { k + 2 }$	$\frac { m + 1 } { m + 4 }$

In the expansion of the polynomial $( 1 + x ) ^ { n }$, the coefficient of $x ^ { 2 }$ is 45. Find the natural number $n$. [3 points]

In the expansion of the polynomial $( x + a ) ^ { 7 }$, when the coefficient of $x ^ { 4 }$ is 280, what is the coefficient of $x ^ { 5 }$? (where $a$ is a constant) [3 points]
(1) 84
(2) 91
(3) 98
(4) 105
(5) 112

In the expansion of the polynomial $( x + a ) ^ { 6 }$, if the coefficient of $x ^ { 4 }$ is 60, what is the value of the positive number $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the coefficient of $x ^ { 4 }$ in the expansion of $\left( x + \frac { 2 } { x } \right) ^ { 8 }$? [3 points]
(1) 108
(2) 112
(3) 116
(4) 120
(5) 124

In the expansion of $\left( x + \frac { 2 } { x } \right) ^ { 8 }$, find the coefficient of $x ^ { 4 }$. [3 points]
(1) 128
(2) 124
(3) 120
(4) 116
(5) 112

What is the coefficient of $x ^ { 4 }$ in the expansion of the polynomial $( 1 + x ) ^ { 7 }$? [3 points]
(1) 42
(2) 35
(3) 28
(4) 21
(5) 14

In the expansion of $\left( 2 x + \frac { 1 } { x ^ { 2 } } \right) ^ { 4 }$, what is the coefficient of $x$? [3 points]
(1) 16
(2) 20
(3) 24
(4) 28
(5) 32

Find the coefficient of $x$ in the expansion of $( 3 x + 1 ) ^ { 8 }$. [3 points]

Find the coefficient of $x ^ { 2 }$ in the expansion of $\left( x + \frac { 3 } { x ^ { 2 } } \right) ^ { 5 }$. [3 points]

In the expansion of the polynomial $( x + 2 ) ^ { 7 }$, what is the coefficient of $x ^ { 5 }$? [2 points]
(1) 42
(2) 56
(3) 70
(4) 84
(5) 98

What is the coefficient of $x ^ { 9 }$ in the expansion of $\left( x ^ { 3 } + 3 \right) ^ { 5 }$? [2 points]
(1) 30
(2) 60
(3) 90
(4) 120
(5) 150

What is the coefficient of $x^{6}$ in the expansion of $\left(x^{3} + 2\right)^{5}$? [2 points]
(1) 40
(2) 50
(3) 60
(4) 70
(5) 80

4. In the expansion of the binomial $( x + 1 ) ^ { n } \left( n \in N _ { + } \right)$, the coefficient of $x ^ { 2 }$ is 15. Then $n =$
A. 4
B. 5
C. 6
D. 7

9. In the expansion of $( 2 + x ) ^ { 5 }$, the coefficient of $x ^ { 3 }$ is $\_\_\_\_$. (Answer with numerals)

11. The coefficient of $x ^ { 2 }$ in the expansion of $( x + 2 ) ^ { 5 }$ equals $\_\_\_\_$. (Answer with a number)

11. In the expansion of $( 2 x - 1 ) ^ { 8 }$, the coefficient of the term containing $x^4$ is $\_\_\_\_$ (answer with a number).