List of top Mathematics Questions on Binomial theorem

If the coefficient of $x^3$ in the binomial expansion of $(2 + x)^n$ is 160, then the coefficient of $x^6$ in the binomial expansion of $(2 - x^2)^n$ is

Let $(2 - x)^9 = a_0 + a_1x + a_2x^2 + \dots + a_9x^9$. Then the value of $a_1 + a_2 + a_3 + \dots + a_8$ is equal to

If the coefficient of $x^4$ in the binomial expansion of $(4x + a)^7$ is -1120, then the value of $a$ is equal to

Let $[t]$ represent the greatest integer less than or equal to $t$. If \[ x = (7\sqrt{5} + 15)^9 \quad \text{and} \quad y = (5\sqrt{7} + 13)^{11}, \] then $[x]$ and $[y]$ are:

If $\alpha_n$ is the coefficient of $x^n$ in the expansion of $(1-x)^{-5}$ and $\beta_n$ is the coefficient of $x^n$ in the expansion of $(1-x)^{-4}$, then $\alpha_{12} + \beta_{13}$ is equal to:

Evaluate binomial sum involving $\sum C_r \frac{2^r}{r+1}$ for $n=10$.}

Approximate $(0.98)^{0.2}$ using binomial expansion.}

If \[ (1+x)^n=C_0+C_1x+C_2x^2+\cdots+C_nx^n \] where \[ C_i={}^{n}C_i, \] then \[ (C_0+C_1)(C_1+C_2)\cdots(C_{n-1}+C_n) \] equals

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 term independent of $x$ in expansion of \[ \left(\frac{\sqrt{x}}{2}-\frac{3}{x}\right)^{12} \] is

Coefficient of $x^3$ in the expansion of \[ \frac{(1-2x^2)^{\frac13}}{(2+x)^{\frac12}} \] is

The term independent of $x$ in expansion of \[ \left(\frac{\sqrt{x}}{2}-\frac{3}{x}\right)^{12} \] is

If the coefficient of $x^2$ in the expansion of \[ (1+x)^5(1-x)^4 \] is $k$, then $k$ is equal to:

If the coefficients of the first, second, and third terms in the expansion of $ (1+x)^n $ are in the ratio $1:20:190$, then $n$ is equal to:

Let $(1+ax)(1-2x)^{3}=\sum_{n=0}^{4}a_{n}x^{n}$, where $a$ is a constant. If $a_{2}=0$, then the value of $a$ is

If the coefficient of $y^{3}$ in the binomial expansion of $\left(2\alpha-\frac{y}{2}\right)^{8}$ is -7, then the value of $\alpha$ is equal to

If $x^{22}$ is in the $(r + 1)^{\text{th}}$ term of the binomial expansion of $(3x^3 - x^2)^9$, then the value of $r$ is equal to

If $ 26 \left( \dfrac{2^3 \cdot \binom{12}{2}}{3} + \dfrac{2^5 \cdot \binom{12}{4}}{5} + \dots + \dfrac{2^{13} \cdot \binom{12}{12}}{13} \right) = 3^{13 - \alpha} $, then find the value of $ \alpha $.

If the coefficients of the middle terms in the binomial expansions of $(1 + \alpha x)^{26}$ and $(1 - \alpha x)^{28}$, $\alpha \neq 0$, are equal, then the value of $\alpha$ is:

In the expansion of \[ \left(9x-\frac{1}{3\sqrt{x}}\right)^{18} \] if coefficient of the term independent of $x$ is $221k$, then the value of $k$ is

The coefficient of $ x^2 $ in the binomial expansion of $ (2x^2 + \frac{1}{x})^{10 $ is:

Find the square of the distance of the point $ (5, 6, 7) $ from the line $ \frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4} $:

The value of $ \int_{0}^{\pi/3} \frac{4 - \cos x \sec^3 x}{\cos^3 x} \, dx $ (or equivalent form) is:

In the expansion of $ (1 + \alpha x)^{26} $ and $ (1 - \alpha x)^{28} $, the coefficient of the middle term is the same, then the value of $ \alpha $ is:

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_________