lean_workbook_plus_29342

Consider having to pick $r$ objects from a total of $n$ . This can be split up into two cases. Either the first object is chosen, or it is not. Now assume the first object is chosen. There are clearly $\binom{n-1}{r-1}$ ways to do this. If the first object isn't chosen, there are $\binom{n-1}{r}$ ways to do this. Another way to count this is to obviously just do $\binom{n}{r}$ . Thus we have that $\binom{n-1}{r-1}+\binom{n-1}{r}=\binom{n}{r},$ and we are done. $\blacksquare$