The number of triplets $(x, y, z)$,where $x, y, z$ are distinct non-negative integers satisfying $x+y+z=15$,is

The remainder of $n^4-2n^3-n^2+2n-26$ when divided by $24$ is:

If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to

The number of natural numbers less than $10000$ which are divisible by $5$ and in which no digit is repeated is:

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 ...... .

In an examination,there are $5$ multiple choice questions with $3$ choices each,out of which exactly one is correct. There are $3$ marks for each correct answer,$-2$ marks for each wrong answer,and $0$ marks if the question is not attempted. The number of ways a student appearing in the examination can get exactly $5$ marks is: