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:

The number of seven-digit integers with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only is:

$A$ man $X$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. The total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men,such that $3$ friends of each of $X$ and $Y$ are invited,is:

The number of $5$-digit odd numbers greater than $40,000$ that can be formed by using the digits $3, 4, 5, 6, 7, 0$ such that at least one of its digits is repeated is:

For which value of $n \in N$,does $n!$ have $13$ trailing zeros?

If the number of circular permutations of $9$ distinct things taken $5$ at a time is $n_1$ and the number of linear permutations of $8$ distinct things taken $4$ at a time is $n_2$,then $\frac{n_1}{n_2}=$