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

A binary number is made up of $16$ bits. The probability of an incorrect bit appearing is $p$ and the errors in different bits are independent of one another. The probability of forming an incorrect number is

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. A fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die, then the probability that $\omega^{I_1}+\omega^{\mathrm{I}_2}+\omega^{\mathrm{I}_3}=0$ is

If $4 -$ digit numbers greater than $5,000$ are randomly formed from the digits
$0,\,1,\,3,\,5,$ and $7,$ what is the probability of forming a number divisible by $5$ when, the repetition of digits is not allowed ?

A box contains $24$ identical balls, of which $12$ are white and $12$ are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the $4^{th}$ time on the $7^{th}$ draw is

The probability that the three cards drawn from a pack of $52$ cards are all red is