$A$ 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 domain of definition of the function $f(x) = \log_{[x + \frac{1}{x}]} |x^2 - x - 6| + ^{16-x}C_{2x-1} + ^{20-3x}P_{2x-5}$ is,where $[x]$ denotes the greatest integer function.

The value of ${ }^{34}C_5 + \sum_{i=0}^4 {}^{38-i}C_4$ is

Statement $I$: The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is ${}^9C_3$.
Statement $II$: The number of ways of choosing $3$ places from $9$ different places is ${}^9C_3$.

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

If $^{n + 1}C_3 = 2 \cdot ^nC_2$,then $n =$