Nine balls are drawn simultaneously from a bag containing $5$ white and $7$ black balls. The probability of drawing $3$ white and $6$ black balls is:

Seven chits are numbered $1$ to $7$. Three are drawn one by one with replacement. The probability that the least number on any selected chit is $5$ is:

Three mangoes and three apples are in a box. If two fruits are chosen at random,the probability that one is a mango and the other is an apple is

$\omega$ is a complex cube root of unity. When an unbiased die is thrown $3$ times, if $\beta_1, \beta_2, \beta_3$ are the numbers appeared on the die, then the probability that $\beta_1, \beta_2$ and $\beta_3$ satisfy $\omega^{\beta_1}+\omega^{\beta_2}=-\omega^{\beta_3}$ is

Four numbers are chosen at random (without replacement) from the set $\{1, 2, 3, ..., 20\}$.
Statement-$1$: The probability that the chosen numbers when arranged in some order will form an $A.P.$ is $\frac{1}{85}$.
Statement-$2$: If the four chosen numbers form an $A.P.$,then the set of all possible values of common difference is $\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6\}$.

Consider the set of all $7$-digit numbers formed by the digits $0, 1, 2, 3, 4, 5, 6$,each chosen exactly once. If a number is randomly drawn from this set,the probability that it is divisible by $4$ is