Number of ways in which $5$  different toys can be distributed among $5$ children if exactly one child do not get any toy

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 $\alpha \beta =$

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

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 ${}^{(n-1)}C_{k-1}$

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