1. The problem asks us to prove the binomial coefficient identity: $$\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}$$ 2. This identity is known as Pascal's Rule or Pascal's Identity. It is fundamental in combinatorics and describes how combinatorial numbers relate. 3. To understand it, recall that $\binom{n}{r}$ counts the number of ways to choose $r$ objects from $n$ objects. 4. Consider a set with $n$ elements. Pick a particular element and call it $x$. 5. The left side, $\binom{n}{r}$, counts all ways to choose $r$ elements from the $n$. 6. On the right side, $\binom{n-1}{r}$ counts the number ways to choose $r$ elements from the remaining $n-1$ elements excluding $x$. 7. The term $\binom{n-1}{r-1}$ counts the ways to choose $r-1$ elements from the $n-1$ elements plus choosing $x$ itself, making a total of $r$ elements. 8. Thus, the total ways to choose $r$ elements either include $x$ or exclude $x$. These two disjoint cases cover all possibilities, so their sum equals the total, proving the identity. Final answer: $$\boxed{\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}}$$