Two numbers $a$ and $b$ are chosen at random from the set of first $30$ natural numbers. The probability that ${a^2} - {b^2}$ is divisible by $3$ is

An ordinary cube has four blank faces, one face marked $2$ another marked $3$. Then the probability of obtaining a total of exactly $12$ in $5$ throws, is

There are $3$ bags $A, B$ & $C$. Bag $A$ contains $1$ Red & $2$ Green balls, bag $B$ contains $2$ Red & $1$ Green balls and bag $C$ contains only one green ball. One ball is drawn from bag $A$ & put into bag $B$ then one ball is drawn from $B$ & put into bag $C$ & finally one ball is drawn from bag $C$ & put into bag $A$. When this operation is completed, probability that bag $A$ contains $2$ Red & $1$ Green balls, is -

Six boys and six girls sit in a row randomly. The probability that the six girls sit together

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

Among $15$ players, $8$ are batsmen and $7$ are bowlers. Find the probability that a team is chosen of $6$ batsmen and $5$ bowlers