$\left(\mathrm{i}\right)$. Number of ways of placing $\text{'}n\text{'}$objects in $k$ bins $k\le n$ ) such that no bin is empty is equal to number of integral solution of 

$x_1+x_2+x_3+...+x_k=n$

which is

$={}^{n-1}C_{k-1}$

$\left(\mathrm{ii}\right)$ Let

$x_1+x_2+x_3+...+x_k=n$

Then, number of positive integral solutions are

$={}^{n-1}C_{k-1}$

$\left(\mathrm{iii}\right)$. Number of ways of placing ' $n$ ' objects in $k$ bins such that at least one bin is non-empty is equal to the number of non-negative integrals solution of

$x_1+x_2+x_3+...+x_k=n$

which is $={}^{n+k-1}C_{k-1}$

$\left(iv\right)$

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

$\left[\because {}^{n}C_k=\frac{n!}{(n-r)!r!}\right]$

$\text{LHS}$

$=\frac{n!}{(n-k)!k!}-\frac{(n-1)!}{(n-k-1)!k!}$

$=\frac{n!}{\left(n-k\right)·\left(n-k-1\right)!·k!}-\frac{\left(n-1\right)!}{\left(n-k-1\right)!·k!}$

$=\frac{n!-\left(n-k\right)\left(n-1\right)!}{\left(n-k\right)·\left(n-k-1\right)!·k!}$

$=\frac{n·\left(n-1\right)!-\left(n-k\right)\left(n-1\right)!}{\left(n-k\right)·\left(n-k-1\right)!·k!}$

$=\frac{\left(n-1\right)!\left[n-\left(n-k\right)\right]}{\left(n-k\right)·\left(n-k-1\right)!·k!}$

$=\frac{k·\left(n-1\right)!}{\left(n-k\right)·\left(n-k-1\right)!·k!}$

$=\frac{k·\left(n-1\right)!}{\left(n-k\right)·\left(n-k-1\right)!·k·\left(k-1\right)!}$

$=\frac{(n-1)!}{(n-k)!(k-1)!}={}^{n-1}C_{k-1}$