The number of ways in which an examiner can assign $30$ marks to $8$ questions,giving not less than $2$ marks to any question,is

Let the number of elements in sets $A$ and $B$ be $5$ and $2$ respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements is:

$A$ student has to answer $10$ out of $13$ questions in an examination,choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student is:

The least value of $n$ such that ${}^{(n-1)}C_6 + {}^{(n-1)}C_7 < {}^{n}C_8$ is

The least value of $n$ such that ${ }^{(n-1)} C_3 + { }^{(n-1)} C_4 > { }^n C_3$ is:

The number of ways $16$ identical cubes,of which $11$ are blue and the rest are red,can be placed in a row so that between any two red cubes there should be at least $2$ blue cubes,is