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:

$A$ linguistic club consists of $6$ girls and $4$ boys. $A$ team of $4$ members is to be selected from this group including the selection of a leader (from among these $4$ members) for the team. If the team has to include at most one boy,the number of ways of selecting the team is

Team $A$ consists of $7$ boys and $n$ girls and Team $B$ has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl,then $n$ is equal to

The digit in the unit place of the number $(183!) + (3^{183})$ is

Out of $11$ consecutive natural numbers,if three numbers are selected at random (without repetition),then the probability that they are in $A.P.$ with a positive common difference is:

Let $m$ be a natural number such that $20000 < m < 60000$ and let $k$ be the sum of all the digits in $m$. Then the number of numbers $m$ for which $k$ is even,is