Two numbers $x$ $\&$ $y$ are chosen at random (without replacement) from the set $\{1, 2, 3, ......, 1000\}$. Then the probability that $|x^4 - y^4|$ is divisible by $5$, is -

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

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

A bag contains tickets numbered from $1$ to $20$. Two tickets are drawn. The probability that both the numbers are prime, is

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of $6$ before $B$ throws a total of $7$ and $B$ wins the game if he throws a total of $7$ before $A$ throws a total of six The game stops as soon as either of the players wins. The probability of $A$ winning the game is 

Two friends $A$ and $B$ have equal number of daughters. There are three cinema tickets which are to be distributed among the daughters of $A$ and $B$. The probability that all the tickets go to daughters of $A$ is $1/20$. The number of daughters each of them have is