Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

If the probability that a randomly chosen $6$-digit number formed by using digits $1$ and $8$ only is a multiple of $21$ is $p$, then $96\;p$ is equal to

A bag contains $8$ red and $7$ black balls. Two balls are drawn at random. The probability that both the balls are of the same colour is

A debate club consists of $6$ girls and $4$ boys. A team of $4$ members is to be selected from this club including the selection of a captain (from among these $4$ memiers) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

In a collection of tentickets, there are two winning tickets. From this collection, five tickets are drawn at random Let $p_1$ and $p_2$ be the probabilities of obtaining one and two winning tickets, respectively. Then $p_1+p_2$ lies in the interval

$n$ cadets have to stand in a row. If all possible permutations are equally likely, then the probability that two particular cadets stand side by side, is