Trouble Identifying Counting Problem Formula? Would this be the correct formula for the counting problem
Partition with Identical Items
$$
        \begin{pmatrix}
        n \\
        r  \\ 
        \end{pmatrix}=  \begin{pmatrix}
        n-1 \\
        r  \\ 
        \end{pmatrix} + \begin{pmatrix}
        n-1 \\
        r-1  \\ 
        \end{pmatrix}  \
$$
Moreover would anyone know a good example that might demonstrate a partition with Identical items? 

For an example:
Suppose you had to choose a committee of 4 people, from a group of 7 people. Call these 7 people A, B, C, D, E, F, and G. Then you can choose this committee in ${7}\choose{4}$ ways. Another way you can calculate this is to split it up into two cases:
Case 1: A is part of the 4-person committee. There are ${6}\choose{3}$ ways to choose the other 3 people from the remaining 6 people to form the committee.
Case 2: A is not part of the 4-person committee. There are ${6}\choose{4}$ ways to choose the 4 people that make up the committee from the remaining 6 people.
We see that Case 1 and Case 2 account for all the 4-person committees once and only once. This is an example showing that:
$${{7}\choose{4}} = {{6}\choose{4}} + {{6}\choose{3}}$$
