The sum of the four-digit even numbers that can be formed with the digits $0, 3, 5, 4$ without repetition is:

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

The number of permutations of the digits $1, 2, 3, ..., 7$ without repetition,which neither contain the string $153$ nor the string $2467$,is $........$.

Let $0 \leq r \leq n$. If ${ }^{n+1} C_{r+1} : { }^{n} C_{r} : { }^{n-1} C_{r-1} = 55 : 35 : 21$,then $2n + 5r$ is equal to:

The largest power of $2$ that divides $\frac{200!}{100!}$ is

$A$ group of students comprises $5$ boys and $n$ girls. If the number of ways,in which a team of $3$ students can be randomly selected from this group such that there is at least one boy and at least one girl in each team,is $1750$,then $n$ is equal to