List of top Mathematics Questions on Binomial theorem asked in MET

If the coefficient of \(x^m\) in the expansion of \(\left(\sqrt{2x} + \sqrt[3]{\frac{3}{x^2}}\right)^9\) is equal to \(k\), then \(k\) is:
If the number of terms in the expansion of \((x\sqrt{180} + \sqrt[3]{432})^{200}\) having integral coefficients is \(n\), then the value of \([n/6]\) is:
If the coefficients of \(x^3\) and \(x^4\) in the expansion of \((1 + ax + bx^2)(1 - 2x)^{18}\) are both zero, then \((a,b)\) is equal to
The coefficient of \(x\) in the expansion of \((1 + x + x^2 + x^3)^{-3\) is}
In the usual notation, \(\frac{^nC_1}{2} + \frac{^nC_2}{3} + \cdots + \frac{^nC_n}{n+1}\) is equal to

The coefficient of \(x^4\) in \((1 + x + x^3 + x^4)^{10}\) is

If the coefficients of the $r$th term and the $(r+1)$th term in the expansion of $(1+x)^{20}$ are in the ratio $1:2$, then $r$ equals
The first three terms in the expansion of $(1 + ax)^{n}$ $(n \neq 0)$ are $1$, $6x$ and $16x^{2}$. Then the values of $a$ and $n$ are respectively
The coefficient of $x^{-9}$ in the expansion of $\left(\dfrac{x^{2}}{2} - \dfrac{2}{x}\right)^{9}$ is}
The middle term of $\left(x - \frac{1}{x}\right)^6$ is
Sum of the last 30 coefficients in the expansion of \[ (1 + x)^{59}, \] when expanded in ascending powers of \( x \), is:
The coefficient of 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:
If \(iz^4 + 1 = 0\) then \(z\) can take the value
A unit vector coplanar with \(\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) and perpendicular to \(\mathbf{i} + \mathbf{j} + \mathbf{k}\) is
The coefficient of \( x^4 \) in the expansion of \( (1 + x + x^2 + x^3)^n \) is
If \( x + y = 1 \), then \[ \sum_{r=0}^{n} r^2 \cdot n C_r x^r y^{n-r} \quad \text{is equal to:} \]
The term independent of \( x \) in the expansion of \( \left[ \sqrt{\frac{x}{3}} + \sqrt{\frac{3}{2}} x^2 \right]^{10} \) is ________.
The greatest coefficient in the expansion of $(1+x)²n$ is ________.
The number of terms in the expansion of $(\sqrt5+\sqrt[4]11)¹24$ which are integers is equal to ________.
The constant term in the expansion of $(1+x)ᵐ(1+\frac1x)ⁿ$ is ________.
The number of terms in the expansion of \( (x + a)^{n} \) is: