If the number of $5$-element subsets of the set $A = \{a_1, a_2, \ldots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is

$100$ cells each of e.m.f. $5 \ V$ and internal resistance $1 \ \Omega$ are to be arranged so as to produce maximum current in a $25 \ \Omega$ resistance. Each row is to contain an equal number of cells. The number of rows should be

Which of the following is a redox reaction?

$\left|\begin{array}{lll}24 & 25 & 26 \\ 25 & 26 & 27 \\ 26 & 27 & 27\end{array}\right|$ is equal to

$\begin{aligned} & A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^\beta\end{array}\right] \\ & \Rightarrow[A(\alpha, \beta)]^{-1}=\end{aligned}$

If $\vec{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\vec b = \frac{1}{7}(2 \hat{i} + 3 \hat{j} - 6 \hat{k})$,then the value of $(2\vec a - \vec b) \cdot [(\vec a \times \vec b) \times (\vec a + 2\vec b)]$ is