The numbers $1, 2, 3, \ldots, n$ are arranged in a random order. The probability that the digits $1, 2, 3, \ldots, k$ appear as a block in that order is:

In an examination,there are $3$ multiple-choice questions,and each question has $4$ choices. The number of ways in which a student can fail to get all answers correct is:

The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} \cdot {}^{23}C_{r}$ is $.......$

$3$ boys $B_i, i = 1, 2, 3$ and $6$ girls $G_i, i = 1, 2, . . . , 6$ are to be seated in a row. The number of ways they can be seated so that $B_1, B_2$ are separated and $G_1, G_2$ are also separated is equal to:

For integers $n$ and $r$,let $\binom{n}{r} = \begin{cases} ^{n}C_{r}, & \text{if } n \geq r \geq 0 \\ 0, & \text{otherwise} \end{cases}$. The maximum value of $k$ for which the sum $\sum_{i=0}^{k}\binom{10}{i}\binom{15}{k-i} + \sum_{i=0}^{k+1}\binom{12}{i}\binom{13}{k+1-i}$ exists,is equal to ...... .

If a number is chosen at random from the four-digit numbers formed by using the digits $0, 1, 2, 3, 4, 6$ without repetition,then the probability that it is divisible by $4$ is