Consider a cube of side length $9$ units. Let $\left(x,y,z\right)$ be coordinates of points on or inside the cube such that $x,y,z\in I$ and $0\le x,y,z\le 9$. If total number of ways of selecting two distinct points among these such that their mid-point is also having integral coordinates is $N$, then which of the following is/are TRUE?

If $\alpha$ and $\beta$ are the greatest divisors of $n\left(n^2-1\right)$ and $2n\left(n^2+2\right)$ respectively for all $n\in N$, then $\alpha \beta =$

Consider the following statements
$\mathrm{i}$. Number of ways of placing $\text{'}n\text{'}$objects in $k$ bins $k\le n$ ) such that no bin is empty is ${}^{(n-1)}C_{k-1}$

$\mathrm{ii}$. Number of ways of writing a positive integer " $n$ ' into a sum of $k$ positive integers is ${}^{(n-1)}C_{k-1}$

$\mathrm{iii}$. Number of ways of placing ' $n$ ' objects in $k$ bins such that at least one bin is non-empty is ${}^{(n-1)}C_{k-1}$

$\mathrm{iv}$. ${}^{n}C_k-{}^{n-1}C_k={}^{(n-1)}C_{k-1}$

If $a$ is the number of all even divisors and $b$ is the number of all odd divisors of the number $10800$, then $2a+3b=$