In the expansion of $(x-1)(x-2) \ldots (x-18)$, the coefficient of $x^{17}$ is:

The coefficients of two consecutive terms in the expansion of $(1 + x)^n$ will be equal,if

Let $a_0, a_1, \ldots, a_{23}$ be real numbers such that $(1+\frac{2}{5} x)^{23} = \sum_{i=0}^{23} a_i x^i$ 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 $....$ .

If the coefficients of the $(2r+1)^{\text{th}}$ term and the $(r+1)^{\text{th}}$ term in the expansion of $(1+x)^{42}$ are equal,then $r$ can be

If the coefficients of the $(2r+1)$-th term and the $(r+2)$-th term in the expansion of $(1+x)^{43}$ are equal,then $r$ is equal to:

If the $k^{\text{th}}$ term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^6$ is independent of $x$,then the numerically greatest term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^k$ when $x = \frac{2}{3}$ is: