If $\frac { 6 } { 3 ^ { 12 } } + \frac { 10 } { 3 ^ { 11 } } + \frac { 20 } { 3 ^ { 10 } } + \frac { 40 } { 3 ^ { 9 } } + \ldots + \frac { 10240 } { 3 } = 2 ^ { n } \cdot m$, where $m$ is odd, then $m \cdot n$ is equal to $\_\_\_\_$.

If the maximum value of the term independent of $t$ in the expansion of $\left( t ^ { 2 } x ^ { \frac { 1 } { 5 } } + \frac { 1 - x ^ { \frac { 1 } { 10 } } } { t } \right)^{10}$, $x \geq 0$, is $K$, then $8K$ is equal to $\_\_\_\_$.

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 $\_\_\_\_$.

Let the coefficients of the middle terms in the expansion of $\left( \frac { 1 } { \sqrt { 6 } } + \beta x \right) ^ { 4 } , ( 1 - 3 \beta x ) ^ { 2 }$ and $\left( 1 - \frac { \beta } { 2 } x \right) ^ { 6 } , \beta > 0$, respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50 - \frac { 2 d } { \beta ^ { 2 } }$ is equal to $\_\_\_\_$.

If $1 + \left( 2 + { } ^ { 49 } C _ { 1 } + { } ^ { 49 } C _ { 2 } + \ldots + { } ^ { 49 } C _ { 49 } \right) \left( { } ^ { 50 } C _ { 2 } + { } ^ { 50 } C _ { 4 } + \ldots + { } ^ { 50 } C _ { 50 } \right)$ is equal to $2 ^ { n } \cdot m$, where $m$ is odd, then $n + m$ is equal to $\_\_\_\_$.

The value of $\sum _ { r } ^ { 22 } = 0 { } ^ { 22 } C _ { r } \cdot { } ^ { 23 } C _ { r }$ is
(1) ${ } ^ { 45 } C _ { 23 }$
(2) ${ } ^ { 44 } C _ { 23 }$
(3) ${ } ^ { 45 } C _ { 24 }$
(4) ${ } ^ { 44 } C _ { 22 }$

If the coefficient of $x^7$ in $\left(ax - \frac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax + \frac{1}{bx^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to:
(1) 11
(2) 44
(3) 22
(4) 33

The value of $\frac{1}{1! \cdot 50!} + \frac{1}{3! \cdot 48!} + \frac{1}{5! \cdot 46!} + \ldots + \frac{1}{49! \cdot 2!} + \frac{1}{51! \cdot 1!}$ is
(1) $\frac{2^{50}}{50!}$
(2) $\frac{2^{50}}{51!}$
(3) $\frac{2^{51}}{51!}$
(4) $\frac{2^{51}}{50!}$

If $\left( { } ^ { 30 } C _ { 1 } \right) ^ { 2 } + 2 \left( { } ^ { 30 } C _ { 2 } \right) ^ { 2 } + 3 \left( { } ^ { 30 } C _ { 3 } \right) ^ { 2 } \ldots\ldots.. 30 \left( { } ^ { 30 } C _ { 30 } \right) ^ { 2 } = \frac { \alpha 60! } { ( 30! ) ^ { 2 } }$, then $\alpha$ is equal to
(1) 30
(2) 60
(3) 15
(4) 10

The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4x}{5} + \frac{5}{2x^2}\right)^9$, is $\_\_\_\_$.

If the coefficients of $x$ and $x ^ { 2 }$ in $( 1 + x ) ^ { p } ( 1 - x ) ^ { q }$ are 4 and $-5$ respectively, then $2p + 3q$ is equal to
(1) 60
(2) 69
(3) 66
(4) 63

The coefficient of $x ^ { 5 }$ in the expansion of $\left( 2 x ^ { 3 } - \frac { 1 } { 3 x ^ { 2 } } \right) ^ { 5 }$ is
(1) $\frac { 80 } { 9 }$
(2) 9
(3) 8
(4) $\frac { 26 } { 3 }$

Let $\left( a + b x + c x ^ { 2 } \right) ^ { 10 } = \sum _ { i = 10 } ^ { 20 } p _ { i } x ^ { i } , a , b , c \in \mathbb { N }$. If $p _ { 1 } = 20$ and $p _ { 2 } = 210$, then $2 ( a + b + c )$ is equal to
(1) 6
(2) 15
(3) 12
(4) 8

Let the sum of the coefficient of first three terms in the expansion of $\left( x - \frac { 3 } { x ^ { 2 } } \right) ^ { n } ; x \neq 0 , n \in N$ be 376. Then, the coefficient of $x ^ { 4 }$ is equal to:

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2} - \frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^\alpha \beta$ where $\beta < 0$ is an odd number, then $|\alpha l - \beta|$ is equal to $\_\_\_\_$.

If $(20)^{19} + 2(21)(20)^{18} + 3(21)^{2}(20)^{17} + \ldots + 20(21)^{19} = k(20)^{19}$, then $k$ is equal to $\_\_\_\_$.

The absolute difference of the coefficients of $x ^ { 10 }$ and $x ^ { 7 }$ in the expansion of $\left( 2 x ^ { 2 } + \frac { 1 } { 2 x } \right) ^ { 11 }$ is equal to
(1) $13 ^ { 3 } - 13$
(2) $11 ^ { 3 } - 11$
(3) $10 ^ { 3 } - 10$
(4) $12 ^ { 3 } - 12$

If $a _ { r }$ is the coefficient of $x ^ { 10 - r }$ in the Binomial expansion of $( 1 + x ) ^ { 10 }$, then $\sum _ { r = 1 } ^ { 10 } r ^ { 3 } \left( \frac { a _ { r } } { a _ { r - 1 } } \right) ^ { 2 }$ is equal to
(1) 4895
(2) 1210
(3) 5445
(4) 3025

Let the coefficients of three consecutive terms in the binomial expansion of $( 1 + 2 x ) ^ { \mathrm { n } }$ be in the ratio $2 : 5 : 8$. Then the coefficient of the term, which is in the middle of these three terms, is

If $\frac { 1 } { n + 1 } { } ^ { n } C _ { n } + \frac { 1 } { n } { } ^ { n } C _ { n - 1 } + \ldots + \frac { 1 } { 2 } { } ^ { n } C _ { 1 } + { } ^ { n } C _ { 0 } = \frac { 1023 } { 10 }$ then $n$ is equal to
(1) 9
(2) 8
(3) 7
(4) 6

If the coefficients of $x^{7}$ in $\left(ax^{2} + \frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax - \frac{1}{3bx^{2}}\right)^{11}$ are equal, then
(1) $729ab = 32$
(2) $32ab = 729$
(3) $64ab = 243$
(4) $243ab = 64$

The constant term in the expansion of $\left( 2 x + \frac { 1 } { x ^ { 7 } } + 3 x ^ { 2 } \right) ^ { 5 }$ is $\_\_\_\_$.

$\sum _ { k = 0 } ^ { 6 } { } ^ { 51 - k } C _ { 3 }$ is equal to
(1) ${ } ^ { 51 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(2) ${ } ^ { 51 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$
(3) ${ } ^ { 52 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(4) ${ } ^ { 52 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$

If the co-efficient of $x ^ { 9 }$ in $\left( \alpha x ^ { 3 } + \frac { 1 } { \beta x } \right) ^ { 11 }$ and the co-efficient of $x ^ { - 9 }$ in $\left( \alpha x - \frac { 1 } { \beta x ^ { 3 } } \right) ^ { 11 }$ are equal, then $( \alpha \beta ) ^ { 2 }$ is equal to

If the coefficients of three consecutive terms in the expansion of $( 1 + x ) ^ { n }$ are in the ratio $1 : 5 : 20$ then the coefficient of the fourth term is
(1) 2436
(2) 5481
(3) 1827
(4) 3654