The coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + 2 x ^ { 2 } \right) ( 1 + x ) ^ { 4 }$ is
A. 12
B. 16
C. 20
D. 24

4. The coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + 2 x ^ { 2 } \right) ( 1 + x ) ^ { 4 }$ is
A. 12
B. 16
C. 20
D. 24

The coefficient of $x ^ { 3 } y ^ { 3 }$ in the expansion of $\left( x + \frac { y ^ { 2 } } { x } \right) ( x + y ) ^ { 5 }$ is
A. 5
B. 10
C. 15
D. 20

13. The coefficient of $x ^ { 2 } y ^ { 6 }$ in the expansion of $\left( 1 - \frac { y } { x } \right) ( x + y ) ^ { 8 }$ is $\_\_\_\_$ (answer with a number).

Coefficient of $x^{11}$ in the expansion of $\left(1 + x^2\right)^4 \left(1 + x^3\right)^7 \left(1 + x^4\right)^{12}$ is
(A) 1051
(B) 1106
(C) 1113
(D) 1120

The coefficient of $x ^ { 9 }$ in the expansion of $( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 3 } \right) \ldots \left( 1 + x ^ { 100 } \right)$ is

The coefficient of $x^{7}$ in the expansion of $\left(1-x-x^{2}+x^{3}\right)^{6}$ is
(1) $-132$
(2) $-144$
(3) $132$
(4) $144$

If $f ( y ) = 1 - ( y - 1 ) + ( y - 1 ) ^ { 2 } - ( y - 1 ) ^ { 3 } + \ldots - ( y - 1 ) ^ { 17 }$ then the coefficient of $y ^ { 2 }$ in it is
(1) ${ } ^ { 17 } \mathrm { C } _ { 2 }$
(2) ${ } ^ { 17 } \mathrm { C } _ { 3 }$
(3) ${ } ^ { 18 } \mathrm { C } _ { 2 }$
(4) ${ } ^ { 18 } \mathrm { C } _ { 3 }$

The middle term in the expansion of $\left( 1 - \frac { 1 } { x } \right) ^ { n } \left( 1 - x ^ { n } \right)$ in powers of $x$ is
(1) ${ } ^ { 2 n } \mathrm { C } _ { n - 1 }$
(2) ${ } ^ { - 2 n } \mathrm { C } _ { n }$
(3) ${ } ^ { 2 n } \mathrm { C } _ { n - 1 }$
(4) ${ } ^ { 2 n } \mathrm { C } _ { n }$

The coefficient of $x ^ { 2 }$ in the expansion of the product $\left( 2 - x ^ { 2 } \right) \left\{ \left( 1 + 2 x + 3 x ^ { 2 } \right) ^ { 6 } + \left( 1 - 4 x ^ { 2 } \right) ^ { 6 } \right\}$ is
(1) 107
(2) 108
(3) 155
(4) 106

The coefficient of $x ^ { 7 }$ in the expression $( 1 + x ) ^ { 10 } + x ( 1 + x ) ^ { 9 } + x ^ { 2 } ( 1 + x ) ^ { 8 } + \ldots + x ^ { 10 }$, is
(1) 210
(2) 330
(3) 120
(4) 420

The coefficient of $x ^ { 256 }$ in the expansion of $( 1 - x ) ^ { 101 } \left( x ^ { 2 } + x + 1 \right) ^ { 100 }$ is:
(1) ${ } ^ { 100 } C _ { 16 }$
(2) ${ } ^ { 100 } C _ { 15 }$
(3) ${ } ^ { - 100 } C _ { 16 }$
(4) ${ } ^ { - 100 } C _ { 15 }$

The coefficient of $x^{101}$ in the expression $(5 + x)^{500} + x(5 + x)^{499} + x^2(5 + x)^{498} + \ldots + x^{500}$, $x > 0$ is
(1) ${}^{501}C_{101} \times 5^{399}$
(2) ${}^{501}C_{101} \times 5^{400}$
(3) ${}^{501}C_{100} \times 5^{400}$
(4) ${}^{500}C_{101} \times 5^{399}$

If the coefficients of $x$ and $x ^ { 2 }$ in the expansion of $( 1 + x ) ^ { p } ( 1 - x ) ^ { q } , p , q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x ^ { 3 }$ is equal to $\_\_\_\_$.

The coefficient of $x^7$ in $(1 - x + 2x^3)^{10}$ is $\_\_\_\_$.

The coefficient of $x ^ { 2012 }$ in the expansion of $1 - x ^ { 2008 } 1 + x + x ^ { 2007 }$ is equal to $\_\_\_\_$ .

$$P ( x ) = ( x + 2 ) ^ { 4 } + 3 ( x + 1 ) ^ { 3 }$$
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

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