Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ By expanding $( 1 + x ) ^ { n }$ for two appropriately chosen real numbers $x$, show that $$\sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } \binom { n } { 2 p } = 2 ^ { n - 1 }.$$

Show that, for all $n \in \mathbb { N } ^ { * }$,
$$\frac { 4 ^ { n } } { 2 n } \leqslant \binom { 2 n } { n } < 4 ^ { n } .$$

The value of $\lim_{n \to \infty} \frac{\sum_{r=0}^{n} {}^{2n}C_{2r} \times 3^{r}}{\sum_{r=0}^{n-1} {}^{2n}C_{2r+1} \times 3^{r}}$ is
(a) 0
(b) 1
(c) $\sqrt{3}$
(d) $(\sqrt{3}-1)/(\sqrt{3}+1)$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is
(A) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(B) $\binom { 6 } { 3 } 3 !$
(C) 33
(D) $3 \binom { 6 } { 3 }$

For $k \geq 1$, the value of $$\binom { n } { 0 } + \binom { n + 1 } { 1 } + \binom { n + 2 } { 2 } + \cdots + \binom { n + k } { k }$$ equals
(A) $\binom { n + k + 1 } { n + k }$
(B) $( n + k + 1 ) \binom { n + k } { n + 1 }$
(C) $\binom { n + k + 1 } { n + 1 }$
(D) $\binom { n + k + 1 } { n }$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is:
(a) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(b) $\frac { 6 ! } { 3 ! }$
(c) 33
(d) $3 \cdot \left( \frac { 6 ! } { 3 ! 3 ! } \right)$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is:
(a) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(b) $\frac { 6 ! } { 3 ! }$
(c) 33
(d) $3 \cdot \left( \frac { 6 ! } { 3 ! 3 ! } \right)$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is
(A) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(B) $\binom { 6 } { 3 } 3 !$
(C) 33
(D) $3 \binom { 6 } { 3 }$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is
(A) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(B) $\binom { 6 } { 3 } 3 !$
(C) 33
(D) $3 \binom { 6 } { 3 }$

The number $$\left( \frac { 2 ^ { 10 } } { 11 } \right) ^ { 11 }$$ is
(A) strictly larger than $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2 \binom{10}{5}$
(B) strictly larger than $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2$ but strictly smaller than $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2 \binom{10}{5}$
(C) less than or equal to $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2$
(D) equal to $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2 \binom{10}{5}$

The number $$\left( \frac { 2 ^ { 10 } } { 11 } \right) ^ { 11 }$$ is
(A) strictly larger than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$
(B) strictly larger than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 }$ but strictly smaller than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$
(C) less than or equal to $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 }$
(D) equal to $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$

Let the integers $a _ { i }$ for $0 \leq i \leq 54$ be defined by the equation
$$\left( 1 + X + X ^ { 2 } \right) ^ { 27 } = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { 54 } X ^ { 54 }$$
Then, $a _ { 0 } + a _ { 3 } + a _ { 6 } + a _ { 9 } + \cdots + a _ { 54 }$ equals
(A) $3 ^ { 26 }$
(B) $3 ^ { 27 }$
(C) $3 ^ { 28 }$
(D) $3 ^ { 29 }$.

Let $n > 1$, and let us arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the first three coefficients are in arithmetic progression. Then, the number of terms where $x$ appears with an integer power, is
(A) 3
(B) 2
(C) 1
(D) 0

For $k \geq 1$, the value of $\binom { n } { 0 } + \binom { n + 1 } { 1 } + \binom { n + 2 } { 2 } + \cdots + \binom { n + k } { k }$ equals
(a) $\binom { n + k + 1 } { n + k }$.
(B) $( n + k + 1 ) \binom { n + k } { n + 1 }$.
(C) $\binom { n + k + 1 } { n + 1 }$.
(D) $\binom { n + k + 1 } { n }$.

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 $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ 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) $\frac { n - 4 } { 5 }$