List of top Mathematics Questions on Binomial theorem

If for $ 3 \leq r \leq 30 $, $ ^{30}C_{30-r} + 3 \left( ^{30}C_{31-r} \right) + 3 \left( ^{30}C_{32-r} \right) + ^{30}C_{33-r} = ^m C_r $, then $ m $ equals to_________

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:

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

If
\[ A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, \]
then the determinant of the matrix $ A^{2025} - 3A^{2024} + A^{2023} $ 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 value of \[ \binom{100}{50} + \binom{100}{51} + \binom{100}{52} + \dots + \binom{100}{100} \] 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 \[ 30\alpha = \underline{\hspace{1cm}} \]

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

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

Numerically greatest term in the expansion of $(3x-4y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is

Let K be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[6]{3})^{6144}$. If the coefficient of $x^P (P \in N)$ in the expansion of $\frac{1}{(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})}$ is $a_P$, then $a_K - a_{K+1} - a_{K-1} =$

When $|x|<1/2$, the coefficient of $x^6$ in the expansion of $(\frac{2-x^2}{1+2x})^6$ is

If $C_0, C_1, C_2, \dots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$ then the value of $\sum r^3 \cdot C_r$ when $n = 5$ is

The coefficient of $x^{12}$ in the expansion of $(x^2+2x+2)^8$ 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 expression $ 5^{2n} - 48n + k $ is divisible by 24 for all $ n \in \mathbb{N} $, then the least positive integral value of k is

If $C_0, C_1, C_2, \dots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$ then $\sum_{r=1}^{n} \frac{r C_r}{C_{r-1}} =$

Numerically greatest term in the expansion of $(2x-3y)^n$ when $x=\frac{7}{5}, y=\frac{3}{7}$ and $n=13$ is

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

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