If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text {th }}$ roll than the number obtained in the $(i-1)^{\text {th }}$ roll, $i=2,3$, is equal to :

Three numbers are chosen at random from $1$ to $15$ . The probability that no two numbers are consecutive, is

From a group of $7$ men and $4$ ladies a committee of $6$ persons is formed, then the probability that the committee contains $2$ ladies is

Let $A$ and $B$ be two finite sets having $m$ and $n$ elements respectively such that $m \le n.\,$ A mapping is selected at random from the set of all mappings from $A$ to $B$. The probability that the mapping selected is an injection is

If $12$ identical balls are to be placed randomly in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is

Let $X$ be a set containing $10$ elements and $P(X)$ be its power set. If $A$ and $B$ are picked up at random from $P(X),$ with replacement, then the probability that $A$ and $B$ have equal number elements, is