16. In the binomial expansion of ( $\mathrm { a } - \mathrm { b } ) \mathrm { n } , \mathrm { n } \geq 5$, the sum of the 5 th and 6 th terms is zero. Then a/b equals:
(A) $( n - 5 ) / 6$
(B) $( n - 4 ) / 5$
(C) $5 / ( n - 4 )$
(D) $6 / ( n - 5 )$

6. The sum
$$\sum _ { i = 0 } ^ { m } \binom { 10 } { i } \binom { 20 } { m - i } \text {, Where } \binom { p } { q } = 0$$
if $p > q$ is maximum when $m$ is
(A) 5
(B) 10
(C) 15
(D) 20

Prove that $$\begin{aligned} & 2 ^ { k } \binom { n } { 0 } \binom { n } { k } - 2 ^ { n - 1 } \binom { n } { 1 } \binom { n - 1 } { k - 1 } + 2 ^ { k - 2 } \\ & \binom { n } { 2 } \binom { n - 2 } { k - 2 } - \ldots \ldots \ldots ( - 1 ) ^ { k } \binom { n } { k } \binom { n - k } { 0 } = \binom { n } { k } . \end{aligned}$$

8.
$$\binom { 30 } { 0 } \binom { 30 } { 10 } - \binom { 30 } { 1 } \binom { 30 } { 11 } + \ldots \ldots \binom { 30 } { 20 } \binom { 30 } { 30 } =$$
(a) $\quad { } ^ { 30 } \mathrm { C } _ { 11 }$
(b) $\quad { } ^ { 60 } \mathrm { C } _ { 10 }$
(c) $\quad { } ^ { 30 } \mathrm { C } _ { 10 }$
(d) $\quad { } ^ { 65 } \mathrm { C } _ { 55 }$

For $\mathrm { r } = 0,1 , \ldots , 10$, let $\mathrm { A } _ { \mathrm { r } } , \mathrm { B } _ { \mathrm { r } }$ and $\mathrm { C } _ { \mathrm { r } }$ denote, respectively, the coefficient of $\mathrm { x } ^ { \mathrm { r } }$ in the expansions of $( 1 + \mathrm { x } ) ^ { 10 } , ( 1 + \mathrm { x } ) ^ { 20 }$ and $( 1 + \mathrm { x } ) ^ { 30 }$. Then
$$\sum _ { r = 1 } ^ { 10 } A _ { r } \left( B _ { 10 } B _ { r } - C _ { 10 } A _ { r } \right)$$
is equal to
A) $\mathrm { B } _ { 10 } - \mathrm { C } _ { 10 }$
B) $\mathrm { A } _ { 10 } \left( \mathrm {~B} _ { 10 } ^ { 2 } - \mathrm { C } _ { 10 } \mathrm {~A} _ { 10 } \right)$
C) O
D) $\mathrm { C } _ { 10 } - \mathrm { B } _ { 10 }$

The coefficients of three consecutive terms of $( 1 + x ) ^ { n + 5 }$ are in the ratio $5 : 10 : 14$. Then $n =$

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

Let
$$X = \left( { } ^ { 10 } C _ { 1 } \right) ^ { 2 } + 2 \left( { } ^ { 10 } C _ { 2 } \right) ^ { 2 } + 3 \left( { } ^ { 10 } C _ { 3 } \right) ^ { 2 } + \cdots + 10 \left( { } ^ { 10 } C _ { 10 } \right) ^ { 2 }$$
where ${ } ^ { 10 } C _ { r } , r \in \{ 1,2 , \cdots , 10 \}$ denote binomial coefficients. Then, the value of $\frac { 1 } { 1430 } X$ is $\_\_\_\_$.

Suppose $$\det\left[\begin{array}{cc}\sum_{k=0}^{n} k & \sum_{k=0}^{n} {}^nC_k k^2 \\ \sum_{k=0}^{n} {}^nC_k & \sum_{k=0}^{n} {}^nC_k 3^k\end{array}\right] = 0$$ holds for some positive integer $n$. Then $\sum_{k=0}^{n} \frac{{}^nC_k}{k+1}$ equals\_\_\_\_

For nonnegative integers $s$ and $r$, let $$\binom{s}{r} = \begin{cases} \frac{s!}{r!(s-r)!} & \text{if } r \leq s \\ 0 & \text{if } r > s \end{cases}$$ For positive integers $m$ and $n$, let $$g(m, n) = \sum_{p=0}^{m+n} \frac{f(m, n, p)}{\binom{n+p}{p}}$$ where for any nonnegative integer $p$, $$f(m, n, p) = \sum_{i=0}^{p} \binom{m}{i}\binom{n+i}{p}\binom{p+n}{p-i}.$$ Then which of the following statements is/are TRUE?
(A) $g(m, n) = g(n, m)$ for all positive integers $m, n$
(B) $g(m, n+1) = g(m+1, n)$ for all positive integers $m, n$
(C) $g(2m, 2n) = 2g(m, n)$ for all positive integers $m, n$
(D) $g(2m, 2n) = (g(m, n))^{2}$ for all positive integers $m, n$

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x ^ { 5 }$ in the expansion of $\left( a x ^ { 2 } + \frac { 70 } { 27 b x } \right) ^ { 4 }$ is equal to the coefficient of $x ^ { - 5 }$ in the expansion of $\left( a x - \frac { 1 } { b x ^ { 2 } } \right) ^ { 7 }$, then the value of $2 b$ is

Let $a _ { 0 } , a _ { 1 } , \ldots , a _ { 23 }$ be real numbers such that
$$\left( 1 + \frac { 2 } { 5 } x \right) ^ { 23 } = \sum _ { i = 0 } ^ { 23 } a _ { i } x ^ { i }$$
for every real number $x$. Let $a _ { r }$ be the largest among the numbers $a _ { j }$ for $0 \leq j \leq 23$. Then the value of $r$ is $\_\_\_\_$.

In the binomial expansion of $( a - b ) ^ { n } , n \geq 5$, the sum of $5 ^ { \text {th } }$ and $6 ^ { \text {th } }$ terms is zero, then $\frac { a } { b }$ equals
(1) $\frac { 5 } { n - 4 }$
(2) $\frac { 6 } { n - 5 }$
(3) $\frac { n - 5 } { 6 }$
(4) $\frac { n - 4 } { 5 }$

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

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 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