For a set of $5$ true or false questions,no student has written all the correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is

$^{80}C_{40}$ is not divisible by -

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

$A$ 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 $...........$.

The remainder obtained when $1! + 2! + 3! + \ldots + 11!$ is divided by $12$ is

If $4$-digit numbers greater than $5,000$ are randomly formed from the digits $0, 1, 3, 5,$ and $7$,what is the probability of forming a number divisible by $5$ when the repetition of digits is not allowed?