Let $x, y, z$ be three non-negative integers such that $x+y+z=10$. The maximum possible value of $xyz+xy+yz+zx$ is

The total number of $5$-digit numbers, formed by using the digits $1, 2, 3, 5, 6, 7$ without repetition, which are multiples of $6$, is

The number of ways in which an arrangement of $4$ letters of the word $PROPORTION$ can be made is

$A$ test containing $3$ objective-type questions is conducted in a class. Each question has $4$ options and only one option is the correct answer. No two students of the class have answered identically,and no student has written all correct answers. If every student has attempted all the questions,then the maximum possible number of students who have written the test is:

The number of positive integral solutions of $abc = 30$ 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 ...... .