$A$ coin is tossed $8$ times. If the probability that exactly $4$ heads appear in the first six tosses and exactly $3$ heads appear in the last five tosses is $p$,then $96p$ is equal to ————

$A$ number $n$ is randomly selected from the set $\{1, 2, 3, \dots, 1000\}$. The probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer is:

An ellipse is inscribed in a circle and a point is chosen at random inside the circle. If the probability that this point lies outside the ellipse is $\frac{2}{3}$,then the eccentricity of the ellipse is $\frac{a\sqrt{b}}{c}$,where $\gcd(a, c) = 1$ and $b$ is a square-free integer. Find the value of $a \cdot b \cdot c$.

$2$ aeroplanes $I$ and $II$ bomb a target in succession. The probabilities of $I$ and $II$ scoring a hit correctly are $0.3$ and $0.2$ respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the $2^{nd}$ plane is

$12$ balls are distributed among $3$ boxes. The probability that the first box will contain exactly $3$ balls is:

If two smallest squares are chosen at random on a chessboard,then the probability of choosing these squares such that they do not have a side in common is: