Let a die be rolled $n$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $\frac{k}{2^{15}}$,then $k$ is equal to:

$A$ test consists of $6$ multiple choice questions,each having $4$ alternative answers of which only one is correct. The number of ways in which a candidate answers all $6$ questions such that exactly $4$ of the answers are correct is:

If the mean and variance of a binomial variate $X$ are $2$ and $1$ respectively,then the probability that $X$ takes a value greater than $1$ is

When two dice are rolled,let $x$ be the probability that the sum of the numbers appearing on the dice is at most $7$. Let $y$ be the probability of getting a sum of $7$ at least once when a pair of dice are rolled $n$ times. In order to have $y > x$,the minimum value of $n$ is

In a workshop,there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$. If the probability that at most two machines will be out of service on the same day is $\left(\frac{3}{4}\right)^{3} k$,then $k$ is equal to

The probability of getting either all heads or all tails for exactly the second time in the $3^{rd}$ trial,if in each trial three coins are tossed,is