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 $\_\_\_\_$ .

Q63. The coefficient of $x ^ { 70 }$ in $x ^ { 2 } ( 1 + x ) ^ { 98 } + x ^ { 3 } ( 1 + x ) ^ { 97 } + x ^ { 4 } ( 1 + x ) ^ { 96 } + \ldots + x ^ { 54 } ( 1 + x ) ^ { 46 }$ is ${ } ^ { 99 } \mathrm { C } _ { \mathrm { p } } - { } ^ { 46 } \mathrm { C } _ { \mathrm { q } }$. Then a possible value of $p + q$ is :
(1) 55
(2) 83
(3) 61
(4) 68

The coefficient of $\mathrm { x } ^ { 48 }$ in
$1 \cdot ( 1 + \mathrm { x } ) + 2 \cdot ( 1 + \mathrm { x } ) ^ { 2 } + 3 \cdot ( 1 + \mathrm { x } ) ^ { 3 } + \ldots + 100 \cdot ( 1 + \mathrm { x } ) ^ { 100 }$ is
(A) $\left( { } ^ { { } ^ { 101 } } \mathrm { C } _ { 46 } \right) - 100$
(B) $\mathbf { 1 0 0 } \left( { } ^ { \mathbf { 1 0 1 } } \mathbf { C } _ { \mathbf { 4 6 } } \right) - { } ^ { \mathbf { 1 0 1 } } \mathbf { C } _ { \mathbf { 4 7 } }$
(C) $\mathbf { 1 0 0 } \left( { } ^ { { } ^ { 101 } } C _ { 49 } \right) - { } ^ { 101 } C _ { 50 }$
(D) ${ } ^ { { } ^ { 101 } } \mathrm { C } _ { 47 } - { } ^ { 101 } \mathrm { C } _ { 46 }$

The sum of coefficients of $\mathbf { x } ^ { \mathbf { 4 9 9 } }$ and $\mathbf { x } ^ { \mathbf { 5 0 0 } }$ in the binomial expansion of $( 1 + x ) ^ { 1000 } + ( 1 + x ) ^ { 999 } ( x ) + x ^ { 2 } ( 1 + x ) ^ { 998 } + \cdots + x ^ { 1000 }$ is (A) ${ } ^ { 1002 } \mathrm { C } _ { 501 }$ lool terms (B) ${ } ^ { 1001 } \mathrm { C } _ { 501 }$ (C) ${ } ^ { 1001 } \mathrm { C } _ { 500 } \quad \gamma = \frac { x } { 1 + x }$ (D) ${ } ^ { 1002 } \mathrm { C } _ { 500 }$

Find the coefficients of $( a + b x ) \cdot ( c + d x )$ in terms of $a , b , c , d$, and explain why $f ( x ) \cdot g ( x ) = g ( x ) \cdot f ( x )$ for all linear polynomials $f ( x )$ and $g ( x )$.

Let $N$ be a whole number with $N > 1$. Find $$( 1 + x ) \cdot ( 1 + 2 x ) \cdot ( 1 + 3 x ) \cdot ( 1 + 4 x ) \cdot \ldots \cdot ( 1 + 2 N x )$$

20. The coefficient of $x ^ { 2 }$ in the expansion of $\left( 4 - x ^ { 2 } \right) \left[ \left( 1 + 2 x + 3 x ^ { 2 } \right) ^ { 6 } - \left( 1 + 4 x ^ { 3 } \right) ^ { 5 } \right]$ is
A 28
B 72
C 78
D 192
E 240
F 310
G 312
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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 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

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