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?

If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$,then the value of $r$ is equal to:

From a collection of $20$ consecutive natural numbers, four are selected such that they are not consecutive. The number of such selections is

Let the eleven letters $A, B, ....., K$ denote an arbitrary permutation of the integers $(1, 2, ....., 11)$. Then,what is the nature of the product $(A - 1)(B - 2)(C - 3) ..... (K - 11)$?

$6$ different letters of an alphabet are given. Words with $4$ letters are formed from these given letters. The number of words which have at least one letter repeated and no two same letters are together is:

There are $10$ pairs of shoes in a cupboard from which $4$ shoes are picked at random. The probability that there is at least one pair is: