An examination has $3$ multiple-choice questions,and each question has $4$ options. If a student passes only if they answer all questions correctly,in how many ways can they fail?

The number of $4$ letter words (with or without meaning) that can be formed from the eleven letters of the word $'EXAMINATION'$ is

$A$ person forgets his $4-$digit $ATM$ pin code. But he remembers that in the code all the digits are different,the greatest digit is $7$ and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is $...........$.

In a relay race there are five teams $A, B, C, D$ and $E$. What is the probability that $A, B$ and $C$ are the first three to finish (in any order)? (Assume that all finishing orders are equally likely.)

$A$ bag contains $n$ white and $n$ black balls. Pairs of balls are drawn at random without replacement successively,until the bag is empty. If the number of ways in which each pair consists of one white and one black ball is $14400$,then $n$ is equal to

If $8 \cdot {}^{7}P_{r} = 7 \cdot {}^{8}P_{r-1}$,what is the value of $r$?