If $^nC_{12} = ^nC_6$,then $^nC_2 = $

$A$ question paper has two sections $A$ and $B$,in which section-$A$ has $8$ questions and section-$B$ has $6$ questions. $A$ student has to answer a total of $10$ questions,choosing at least $4$ questions from section-$A$ and at least $3$ questions from section-$B$. The number of ways a student can answer the paper is:

In a shop,there are $5$ types of ice-creams available. $A$ child buys $6$ ice-creams.
$Statement-1$: The number of different ways the child can buy the $6$ ice-creams is $^{10}C_5$.
$Statement-2$: The number of different ways the child can buy the $6$ ice-creams is equal to the number of different ways of arranging $6$ $A$'s and $4$ $B$'s in a row.

Let $S_r = \{(x, y, z) : x + y + z = 11, x \geq r, y \geq r, z \geq r, x, y, z, r \in \mathbb{Z}\}$ and $n(S_r)$ represents the number of elements in $S_r$. Then $n(S_2) + n(S_3) + n(S_4) = $

From a class of $25$ students,$10$ are to be chosen for an excursion party. There are $3$ students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

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.