There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, is a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

An unbiased die with faces marked $1, 2, 3, 4, 5$ and $6$ is rolled four times. Out of four face values obtained the probability that the minimum face value is not less than $2$ and the maximum face value is not greater than $5$, 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

In a lottery, a person choses six different natural numbers at random from $1$ to $20$ , and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? [ Hint order of the numbers is not important.]

A word consists of $11$ letters in which there are $7$ consonants and $4$ vowels. If $2$ letters are chosen at random, then the probability that all of them are consonants, is

It is $5 : 2$ against a husband who is $65$ years old living till he is $85$ and $4 : 3$ against his wife who is now $58$, living till she is $78$. If the probability that atleast one of them will be alive for $20$ years, is $'k'$, then the value of $'49k'$ -