$A$ fair coin is tossed a fixed number of times. If the probability of getting five heads is equal to that of getting seven heads, then the probability of getting four heads is

If $X \sim B(7, p)$ is a binomial variate and $P(X=3)=P(X=5)$ then $p=$

$A$ box contains $15$ green and $10$ yellow balls. If $10$ balls are randomly drawn,one by one,with replacement,then the variance of the number of green balls drawn is

For $k > 0$,calculate the value of $\sum_{x=0}^{\infty} \frac{k^x}{x !} \lim _{n}$ ${\rightarrow \infty} \frac{n !}{(n-x) !}\left(1-\frac{k}{n}\right)^{n-x}\left(\frac{1}{n}\right)^x$.

In a multiple-choice test consisting of $8$ questions,each question has four options. For each of the questions,exactly one of the four options is the right answer. $A$ student answers all the questions by choosing one option for each question. The number of ways in which the student can get exactly $5$ correct answers is

$A$ multiple choice test consists of $5$ questions,each question having $4$ responses. There is only one correct response and the remaining $3$ are incorrect responses. If a candidate attempts all the $5$ questions,then the probability that he answers at least $3$ questions incorrectly is