The sum and product of the mean and variance of a binomial distribution are $82.5$ and $1350$ respectively. The number of trials in the binomial distribution is:

Two dice are thrown $5$ times,and each time the sum of the numbers obtained being $5$ is considered a success. If the probability of having at least $4$ successes is $\frac{k}{3^{11}}$,then $k$ is equal to

$A$ random variable $X$ follows the binomial distribution and $X \sim B(n, 0.3)$. If the mean of $X$ is three times as large as the standard deviation of $X$,then $n=$

The minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than $99\%$ is:

If $A$ and $B$ throw two dice $100$ times each simultaneously,then the probability that both of them will get an even number as the total score at the same time in all the throws is

In an examination,there are $10$ true-false type questions. Out of $10$,a student can guess the answer of $4$ questions correctly with probability $\frac{3}{4}$ and the remaining $6$ questions correctly with probability $\frac{1}{4}$. If the probability that the student guesses the answers of exactly $8$ questions correctly out of $10$ is $\frac{27 k}{4^{10}}$,then $k$ is equal to