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

The 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 $a _ { r }$ 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 $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $( 1 + x ) ^ { 99 }$. Let a be the middle term in the expansion of $\left( 2 + \frac { 1 } { \sqrt { 2 } } \right) ^ { 200 }$. If $\frac { { } ^ { 200 } C _ { 99 } K } { a } = \frac { 2 ^ { l } m } { n }$, where $m$ and $n$ are odd numbers, then the ordered pair $( l , \mathrm { n } )$ is equal to: (1) $( 50,51 )$ (2) $( 51,99 )$ (3) $( 50,101 )$ (4) $( 51,101 )$

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

The sum, of the coefficients of the first 50 terms in the binomial expansion of $( 1 - x ) ^ { 100 }$, is equal to
(1) ${ } ^ { 101 } C _ { 50 }$
(2) ${ } ^ { 99 } C _ { 49 }$
(3) $- { } ^ { 101 } C _ { 50 }$
(4) $- { } ^ { 99 } C _ { 49 }$

If $A$ denotes the sum of all the coefficients in the expansion of $\left( 1 - 3 x + 10 x ^ { 2 } \right) ^ { n }$ and $B$ denotes the sum of all the coefficients in the expansion of $\left( 1 + x ^ { 2 } \right) ^ { n }$, then:
(1) $\mathrm { A } = \mathrm { B } ^ { 3 }$
(2) $3 \mathrm {~A} = \mathrm { B }$
(3) $\mathrm { B } = \mathrm { A } ^ { 3 }$
(4) $\mathrm { A } = 3 \mathrm {~B}$

If $\frac { { } ^ { 11 } C _ { 1 } } { 2 } + \frac { { } ^ { 11 } C _ { 2 } } { 3 } + \ldots . . + \frac { { } ^ { 11 } C _ { 9 } } { 10 } = \frac { n } { m }$ with $\operatorname { gcd } ( n , m ) = 1$, then $n + m$ is equal to

Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3, 7, 11, \ldots$, if $40 < \frac{6}{n(n+1)}\sum_{k=1}^{n} S_k < 42$, then $n$ equals $\underline{\hspace{1cm}}$.

Let $\alpha = \sum _ { r = 0 } ^ { n } \left( 4 r ^ { 2 } + 2 r + 1 \right) ^ { n } C _ { r }$ and $\beta = \left( \sum _ { r = 0 } ^ { n } \frac { { } ^ { n } C _ { r } } { r + 1 } \right) + \frac { 1 } { n + 1 }$. If $140 < \frac { 2 \alpha } { \beta } < 281$, then the value of $n$ is $\_\_\_\_$

If $\mathrm { S } ( x ) = ( 1 + x ) + 2 ( 1 + x ) ^ { 2 } + 3 ( 1 + x ) ^ { 3 } + \cdots + 60 ( 1 + x ) ^ { 60 } , x \neq 0$, and $( 60 ) ^ { 2 } \mathrm {~S} ( 60 ) = \mathrm { a } ( \mathrm { b } ) ^ { \mathrm { b } } + \mathrm { b }$, where $a , b \in N$, then $( a + b )$ equal to $\_\_\_\_$

If $\sum _ { r = 0 } ^ { 5 } \frac { { } ^ { 11 } C _ { 2r } } { 2 r + 2 } = \frac { \mathrm { m } } { \mathrm { n } } , \operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } - \mathrm { n }$ is equal to $\_\_\_\_$

If $\sum _ { r = 1 } ^ { 30 } \frac { r ^ { 2 } \left( { } ^ { 30 } C _ { r } \right) ^ { 2 } } { { } ^ { 30 } C _ { r - 1 } } = \alpha \times 2 ^ { 29 }$, then $\alpha$ is equal to $\_\_\_\_$

If $\alpha = 1 + \sum _ { r = 1 } ^ { 6 } ( - 3 ) ^ { r - 1 } \quad { } ^ { 12 } \mathrm { C } _ { 2 r - 1 }$, then the distance of the point $( 12 , \sqrt { 3 } )$ from the line $\alpha x - \sqrt { 3 } y + 1 = 0$ is $\_\_\_\_$.