The sum of the series ${ } ^ { 20 } \mathrm { C } _ { 0 } - { } ^ { 20 } \mathrm { C } _ { 1 } + { } ^ { 20 } \mathrm { C } _ { 2 } - { } ^ { 20 } \mathrm { C } _ { 3 } + \ldots - \ldots + { } ^ { 20 } \mathrm { C } _ { 10 }$ is
(1) $- { } ^ { 20 } \mathrm { C } _ { 10 }$
(2) $\frac { 1 } { 2 } { } ^ { 20 } \mathrm { C } _ { 10 }$
(3) 0
(4) ${ } ^ { 20 } \mathrm { C } _ { 10 }$

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) $144$

The number of terms in the expansion of $\left(y^{1/5} + x^{1/10}\right)^{55}$, in which powers of $x$ and $y$ are free from radical signs are
(1) six
(2) twelve
(3) seven
(4) five

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) ${ } ^ { 2 n } \mathrm { C } _ { n }$

If $n$ is a positive integer, then $(\sqrt{3}+1)^{2n} - (\sqrt{3}-1)^{2n}$ is
(1) an irrational number
(2) an odd positive integer
(3) an even positive integer
(4) a rational number other than positive integers

The sum of the rational terms in the binomial expansion of $\left( 2 ^ { \frac { 1 } { 2 } } + 3 ^ { \frac { 1 } { 5 } } \right) ^ { 10 }$ is :
(1) 25
(2) 32
(3) 9
(4) 41

The ratio of the coefficient of $x ^ { 15 }$ to the term independent of $x$ in the expansion of $\left( x ^ { 2 } + \frac { 2 } { x } \right) ^ { 15 }$ is:
(1) $7 : 16$
(2) $7 : 64$
(3) $1 : 4$
(4) $1 : 32$

If for positive integers $r > 1 , n > 2$, the coefficients of the $( 3r ) ^ { \text {th} }$ and $( r + 2 ) ^ { \text {th} }$ powers of $x$ in the expansion of $( 1 + x ) ^ { 2n }$ are equal, then $n$ is equal to:
(1) $2r + 1$
(2) $2r - 1$
(3) $3r$
(4) $r + 1$

The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right)^{10}$ is
(1) 210
(2) 310
(3) 4
(4) 120

If the coefficients of $x ^ { 3 }$ and $x ^ { 4 }$ in the expansion of $\left( 1 + a x + b x ^ { 2 } \right) ( 1 - 2 x ) ^ { 18 }$ in powers of $x$ are both zero, then $( a , b )$ is equal to
(1) $\left( 14 , \frac { 272 } { 3 } \right)$
(2) $\left( 16 , \frac { 272 } { 3 } \right)$
(3) $\left( 16 , \frac { 251 } { 3 } \right)$
(4) $\left( 14 , \frac { 251 } { 3 } \right)$

The coefficient of $x ^ { 1012 }$ in the expansion of $\left( 1 + x ^ { n } + x ^ { 253 } \right) ^ { 10 }$, (where $n \leq 22$ is any positive integer), is
(1) ${ } ^ { 253 } C _ { 4 }$
(2) ${ } ^ { 10 } C _ { 4 }$
(3) $4 n$
(4) 1

The number of terms in the expansion of $( 1 + x ) ^ { 101 } \left( 1 - x + x ^ { 2 } \right) ^ { 100 }$ in powers of $x$ is
(1) 301
(2) 302
(3) 101
(4) 202

The sum of coefficients of integral powers of $x$ in the binomial expansion of $(1 - 2\sqrt{x})^{50}$ is:
(1) $\frac{1}{2}(3^{50} + 1)$
(2) $\frac{1}{2}(3^{50})$
(3) $\frac{1}{2}(3^{50} - 1)$
(4) $\frac{1}{2}(2^{50} + 1)$

If $A$ and $B$ are coefficients of $x^n$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively, then $\frac{A}{B}$ equals:
(1) $1$
(2) $2$
(3) $\frac{1}{2}$
(4) $\frac{1}{n}$

If the number of terms in the expansion of $\left(1 - \frac{2}{x} + \frac{4}{x^2}\right)^n$, $x \neq 0$, is 28, then the sum of the coefficients of all the terms in this expansion, is:
(1) 64
(2) 2187
(3) 243
(4) 729

The sum $\sum_{r=1}^{9} \frac{10!}{r!(10-r)!}$ is equal to:
(1) $2^{10} - 2$
(2) $2^{10} - 1$
(3) $2^9$
(4) $2^{10}$

If the coefficients of $x ^ { - 2 }$ and $x ^ { - 4 }$, in the expansion of $\left( x ^ { \frac { 1 } { 3 } } + \frac { 1 } { 2 x ^ { \frac { 1 } { 3 } } } \right) ^ { 18 } , ( x > 0 )$, are $m$ and $n$ respectively, then $\frac { m } { n }$ is equal to
(1) 27
(2) 182
(3) $\frac { 5 } { 4 }$
(4) $\frac { 4 } { 5 }$

The value of ${}^{21}C_1 - {}^{10}C_1 + {}^{21}C_2 - {}^{10}C_2 + {}^{21}C_3 - {}^{10}C_3 + {}^{21}C_4 - {}^{10}C_4 + \ldots + {}^{21}C_{10} - {}^{10}C_{10}$ is
(1) $2^{21} - 2^{11}$
(2) $2^{21} - 2^{10}$
(3) $2^{20} - 2^{9}$
(4) $2^{20} - 2^{10}$

The sum of the co-efficient of all odd degree terms in the expansion of $\left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , ( x > 1 )$ is
(1) 2
(2) - 1
(3) 0
(4) 1

If $n$ is the degree of the polynomial,
$$\left[ \frac { 1 } { \sqrt { 5 x ^ { 3 } + 1 } - \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 } + \left[ \frac { 1 } { \sqrt { 5 x ^ { 3 } + 1 } + \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 }$$
and $m$ is the coefficient of $x ^ { n }$ in it, then the ordered pair ( $n , m$ ) is equal to
(1) $\left( 12 , ( 20 ) ^ { 4 } \right)$
(2) $\left( 8,5 ( 10 ) ^ { 4 } \right)$
(3) $\left( 24 , ( 10 ) ^ { 8 } \right)$
(4) $\left( 12,8 ( 10 ) ^ { 4 } \right)$

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

If $n$ is the degree of the polynomial, $\left[ \frac { 2 } { \sqrt { 5 x ^ { 3 } + 1 } - \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 } + \left[ \frac { 2 } { \sqrt { 5 x ^ { 3 } + 1 } + \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 }$ and $m$ is the coefficient of $x ^ { n }$ in it, then the ordered pair $( n , m )$ is equal to
(1) $\left( 8,5 ( 10 ) ^ { 4 } \right)$
(2) $\left( 12,8 ( 10 ) ^ { 4 } \right)$
(3) $\left( 12 , ( 20 ) ^ { 4 } \right)$
(4) $\left( 24 , ( 10 ) ^ { 8 } \right)$

The positive value of $\lambda$ for which the co-efficient of $x ^ { 2 }$ in the expansion $x ^ { 2 } \left( \sqrt { x } + \frac { \lambda } { x ^ { 2 } } \right) ^ { 10 }$ is 720, is
(1) $\sqrt { 5 }$
(2) 3
(3) 4
(4) $2 \sqrt { 2 }$

If $\sum _ { i = 1 } ^ { 20 } \left( \frac { { } ^ { 20 } C _ { i - 1 } } { { } ^ { 20 } C _ { i } + { } ^ { 20 } C _ { i - 1 } } \right) ^ { 3 } = \frac { k } { 21 }$, then $k$ equals
(1) 200
(2) 100
(3) 50
(4) 400