List of top Mathematics Questions on Binomial theorem

The sum of the coefficients of $x^{499}$ and $x^{500}$ in \[ (1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\cdots+x^{1000} \] is:

Let $ C_r $ denote the coefficient of $ x^r $ in the binomial expansion of $ (1+x)^n $, where $ n \in \mathbb{N} $ and $ 0 \le r \le n $. If \[ P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \cdots + \frac{(-2)^n}{n+1} C_n, \] then the value of \[ \sum_{n=1}^{25} \frac{1}{2n} P_n \] equals

The coefficient of $x^{48}$ in \[ (1+x) + 2(1+x)^2 + 3(1+x)^3 + \cdots + 100(1+x)^{100} \] is equal to

If
\[ A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, \]
then the determinant of the matrix $ A^{2025} - 3A^{2024} + A^{2023} $ is

\[ \left( \frac{1}{{}^{15}C_0} + \frac{1}{{}^{15}C_1} \right) \left( \frac{1}{{}^{15}C_1} + \frac{1}{{}^{15}C_2} \right) \cdots \left( \frac{1}{{}^{15}C_{12}} + \frac{1}{{}^{15}C_{13}} \right) = \frac{\alpha^{13}}{{}^{14}C_0 \, {}^{14}C_1 \cdots {}^{14}C_{12}} \]

Then \[ 30\alpha = \underline{\hspace{1cm}} \]

If the coefficient of $ x $ in the expansion of \[ (ax^2 + bx + c)(1 - 2x)^{26} \] is $-56$ and the coefficients of $ x^2 $ and $ x^3 $ are both zero, then $ a + b + c $ is equal to

The value of \[ \frac{{}^{100}C_{50}}{51} + \frac{{}^{100}C_{51}}{52} + \cdots + \frac{{}^{100}C_{100}}{101} \] is:

The coefficient of x$^{48}$ in $1(1+x)+2(1+x)^2+3(1+x)^3 +.....+100(1+x)^{100}$ is:

The value of \[ \binom{100}{50} + \binom{100}{51} + \binom{100}{52} + \dots + \binom{100}{100} \] is:

In the binomial expansion of $ (ax^2 + bx + c)(1 - 2x)^{26} $, the coefficients of $ x, x^2 $, and $ x^3 $ are -56, 0, and 0 respectively. Then, the value of $ (a + b + c) $ is

If in the expansion of $ (1 + x^2)^2(1 + x)^n $, the coefficients of $ x $, $ x^2 $, and $ x^3 $ are in arithmetic progression, then the sum of all possible values of $ n $ (where $ n \geq 3 $) is:

The coefficient of $x^{48}$ in \[ 1(1+x) + 2(1+x)^2 + 3(1+x)^3 + \cdots + 100(1+x)^{100} \] is

The coefficient of $ x^{48} $ in the expansion of \[ 1 + (1+x) + 2(1+x)^2 + 3(1+x)^3 + \dots + 100(1+x)^{100} \] is

If $ \sum_{r=0}^5 \frac{{}^{11}C_{2r+1}}{2r+2} = \frac{m}{n} $, gcd(m, n) = 1, then $ m - n $ is equal to:

The sum of all rational numbers in $ (2 + \sqrt{3})^8 $ is:

The term independent of $ x $ in the expansion of $$ \left( \frac{x + 1}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{x + 1}{x - \sqrt{x}} \right)^{10} $$ for $ x>1 $ is:

If the sum, $ \sum_{r=1}^9 \frac{r+3}{2r} \cdot \binom{9}{r} $ is equal to $ \alpha \cdot \left( \frac{3}{2} \right)^9 - \beta $, then the value of $ (\alpha + \beta)^2 $ is equal to:

Let $ (1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + ... + a_{20} x^{20} $. If $ (a_1 + a_3 + a_5 + ... + a_{19}) - 11a_2 = 121k $, then k is equal to _______

If $ \sum_{r=1}^{9} \left( \frac{r+3}{2^r} \right) \cdot {^9C_r} = \alpha \left( \frac{3}{2} \right)^9 - \beta $, $ \alpha, \beta \in \mathbb{N} $, then $ (\alpha + \beta)^2 $ is equal to

If
$ 2^m 3^n 5^k, \text{ where } m, n, k \in \mathbb{N}, \text{ then } m + n + k \text{ is equal to:} $

The sum of all rational terms in the expansion of $ \left( 2 + \sqrt{3} \right)^8 $ is

The number of rational terms in the binomial expansion of $\left( \frac{1}{2} + \frac{1}{8} \right)^{1016}$ is

If $-3x + 17<-13$, then

In the expansion of \[ \left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n , \, n \in \mathbb{N}, \] if the ratio of the 15^th term from the beginning to the 15^th term from the end is \[ \frac{1}{6}, \] then the value of \[ {}^nC_3 \] is:

For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n-3}$ is 16, then the distance of the point $P(2n-1, n^2-4n)$ from the line $x + y = 8$ is: