The identity given in the image is a well-known combinatorial identity, often referred to as Vandermonde's identity. It can be stated as:

$\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r}$

This identity can be verified using the block-walking interpretation. Here's how we can approach it:

Block-Walking Interpretation

Imagine that you have two grids:

1. A grid of size $m \times n$ represents all possible paths from the bottom-left corner to the top-right corner of the combined grid.
2. A block-walking problem involves counting how many ways we can walk on these grids using only "up" and "right" moves.

• The left side of the identity counts the number of ways to reach a specific point after a certain number of steps. First, we choose $k$ steps to move right from $m$ available "right" moves, and $r-k$ steps to move up from $n$ available "up" moves.
• The right side of the identity simply counts how many ways we can choose $r$ steps from the total available steps, which is the sum of $m$ and $n$ moves.

Intuition:

• Each term $\binom{m}{k} \binom{n}{r-k}$ counts the number of ways to make $k$ right moves from $m$ available and $r-k$ up moves from $n$ available. Summing over all possible values of $k$ gives all possible paths that reach a certain point after exactly $r$ steps.
• The right-hand side $\binom{m+n}{r}$ directly counts the number of ways to take $r$ steps (out of the total $m+n$) with no restriction on which are right and which are up.

Conclusion:

The left side and the right side both count the same thing, but they do so in different ways. Therefore, the identity holds.

Would you like more details or further clarification on any of these steps?

Here are five related questions to explore:

1. How does Vandermonde's identity relate to the binomial theorem?
2. Can you derive the identity using generating functions?
3. How would this identity be applied in the context of probability theory?
4. Can this identity be extended to multinomial coefficients?
5. What other combinatorial identities can be interpreted through block-walking?

Tip: When working with combinatorial identities, it's helpful to interpret sums and products as counting methods on paths or objects, making it easier to understand complex relationships.