If N denotes the number of ways of selecting r objects out of n distinct objects (r $\ge$n) with unlimited repetition but
with each object included at least once in selection, then N is equal to 

Let x_i(l $\le$ i $\le$ n)be the number of objects selected of the i th type. 
Since each object is to be selected at least once,

 we must have x_i$\le$1 and ${\mathrm{x}}_1+{\mathrm{x}}_2+\cdots +{\mathrm{x}}_{\mathrm{n}}=\mathrm{r}$. 
We have to find the number of positive integral solutions of the above equation.
Total number of such solutions is ${ }^{\mathrm{r}-1}{\mathrm{C}}_{\mathrm{n}-1}{=}^{\mathrm{r}-1}{\mathrm{C}}_{\mathrm{r}-\mathrm{n}}$