There are $4$ men and $5$ women in Group $A$,and $5$ men and $4$ women in Group $B$. If $4$ persons are selected from each group,then the number of ways of selecting $4$ men and $4$ women is:

Let $m = (9n^2 + 54n + 80)(9n^2 + 45n + 54)(9n^2 + 36n + 35)$. The greatest positive integer which divides $m$ for all positive integers $n$ 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 $...........$.

All possible $5$-digit numbers,each having $5$ distinct digits,are formed using the digits $\{1, 2, 3, 5, 6, 8\}$. Among them,the number of such numbers which are divisible by $3$ but not by $6$ is:

In an admission test,there are $15$ multiple-choice questions. Each question has $4$ alternatives. For each question,one or more than one answer can be correct. If a student attempts all $15$ questions and marks the answers randomly,in how many different ways can the student answer the question paper?

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