The lowest integer which is greater than $\left( 1 + \frac { 1 } { 10 ^ { 100 } } \right) ^ { 10 ^ { 100 } }$ is
(1) 3
(2) 4
(3) 2
(4) 1

If ${ } ^ { 20 } \mathrm { C } _ { \mathrm { r } }$ is the co-efficient of $x ^ { \mathrm { r } }$ in the expansion of $( 1 + x ) ^ { 20 }$, then the value of $\sum _ { \mathrm { r } = 0 } ^ { 20 } \mathrm { r } ^ { 2 } \left( { } ^ { 20 } \mathrm { C } _ { \mathrm { r } } \right)$ is equal to:
(1) $420 \times 2 ^ { 18 }$
(2) $380 \times 2 ^ { 18 }$
(3) $380 \times 2^{19}$
(4) $420 \times 2 ^ { 19 }$

Let $[ x ]$ denote greatest integer less than or equal to $x$. If for $n \in N , \left( 1 - x + x ^ { 3 } \right) ^ { n } = \sum _ { j = 0 } ^ { 3 n } a _ { j } x ^ { j }$, then $\sum _ { j = 0 } ^ { \left[ \frac { 3 n } { 2 } \right] } a _ { 2 j } + 4 \sum _ { j = 0 } ^ { \left[ \frac { 3 n - 1 } { 2 } \right] } a _ { 2 j + 1 }$ is equal to :
(1) 2
(2) $2 ^ { n - 1 }$
(3) 1
(4) $n$

The value of $-{ } ^ { 15 } C _ { 1 } + 2 \cdot { } ^ { 15 } C _ { 2 } - 3 \cdot { } ^ { 15 } C _ { 3 } + \ldots - 15 \cdot { } ^ { 15 } C _ { 15 } + { } ^ { 14 } C _ { 1 } + { } ^ { 14 } C _ { 3 } + { } ^ { 14 } C _ { 5 } + \ldots + { } ^ { 14 } C _ { 11 }$ is equal to
(1) $2 ^ { 14 }$
(2) $2 ^ { 13 } - 13$
(3) $2 ^ { 16 } - 1$
(4) $2 ^ { 13 } - 14$

Let $\left( 1 + x + 2 x ^ { 2 } \right) ^ { 20 } = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \ldots + a _ { 40 } x ^ { 40 }$, then $a _ { 1 } + a _ { 3 } + a _ { 5 } + \ldots + a _ { 37 }$ is equal to
(1) $2 ^ { 20 } \left( 2 ^ { 20 } - 21 \right)$
(2) $2 ^ { 19 } \left( 2 ^ { 20 } - 21 \right)$
(3) $2 ^ { 19 } \left( 2 ^ { 20 } + 21 \right)$
(4) $2 ^ { 20 } \left( 2 ^ { 20 } + 21 \right)$

Let the coefficients of third, fourth and fifth terms in the expansion of $\left( x + \frac { a } { x ^ { 2 } } \right) ^ { n } , x \neq 0$, be in the ratio $12 : 8 : 3$. Then the term independent of $x$ in the expansion, is equal to $\_\_\_\_$ .

If the constant term in the expansion of $\left( 3 x ^ { 3 } - 2 x ^ { 2 } + \frac { 5 } { x ^ { 5 } } \right) ^ { 10 }$ is $2 ^ { k } \cdot l$, where $l$ is an odd integer, then the value of $k$ is equal to
(1) 6
(2) 7
(3) 8
(4) 9

$\sum _ { i , j = 0 , i \neq j } ^ { n } { } ^ { n } C _ { i } { } ^ { n } C _ { j }$ is equal to
(1) $2 ^ { 2 n } - { } ^ { 2 n } C _ { n }$
(2) $2 ^ { 2 n - 1 } - { } ^ { 2 n - 1 } C _ { n - 1 }$
(3) $2 ^ { 2 n } - \frac { 1 } { 2 } { } ^ { 2 n } C _ { n }$
(4) $2 ^ { n - 1 } + { } ^ { 2 n - 1 } C _ { n }$

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 sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left( x ^ { n } + \frac { 2 } { x ^ { 5 } } \right) ^ { 7 }$ is 939, then the sum of all the possible integral values of $n$ is

Let for the $9 ^ { \text {th } }$ term in the binomial expansion of $( 3 + 6 x ) ^ { n }$, in the increasing powers of $6 x$, to be the greatest for $x = \frac { 3 } { 2 }$, the least value of $n$ is $n _ { 0 }$. If $k$ is the ratio of the coefficient of $x ^ { 6 }$ to the coefficient of $x ^ { 3 }$, then $k + n _ { 0 }$ 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 $\_\_\_\_$.