Three persons enter in a lift at the ground floor. The lift will go up to 10^th floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ________.

If the distance of the point $P(43, \alpha, \beta)$, $\beta < 0$, from the line $\vec{r} = 4\hat{i} - \hat{k} + \mu (2\hat{i} + 3\hat{k}), \mu \in \mathbb{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^2 + \beta^2$ is equal to ________

Let $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B$ be two matrices such that $A^{100} = 100B + I$. Then the sum of all the elements of $B^{100}$ is _______

Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int\limits_0^x e^{(x-t)} f(t)\,dt$, $x \in \mathbb{R}$ and let

$g(x) = \int\limits_0^x (f(t) + 2)^{15} (t - 4)^6 (t + 12)^{17}\,dt$, $x \in \mathbb{R}$.

If $p$ and $q$ are respectively the points of local minima and local maxima of $g$, then the value of $|p+q|$ is equal to ________.