The problem requires proving the following identity involving binomial coefficients:

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

Proof Using Combinatorial Argument:

This identity is often referred to as the "Vandermonde Convolution" and can be proven using a combinatorial argument.

Step-by-step Proof:

1. Combinatorial Interpretation:

   • The left-hand side, $\binom{m+n}{r}$, represents the number of ways to choose $r$ elements from a set of $m+n$ elements.
   • Consider the set as the union of two disjoint subsets: one of size $m$ and the other of size $n$.

2. Breaking Down the Selection:

   • To choose $r$ elements from $m+n$, we can break it down into choosing $k$ elements from the first subset (size $m$) and the remaining $r-k$ elements from the second subset (size $n$).

3. Counting the Ways:

   • The number of ways to choose $k$ elements from the first subset is given by $\binom{m}{k}$.
   • The number of ways to choose $r-k$ elements from the second subset is given by $\binom{n}{r-k}$.

4. Applying the Multiplication Principle:

   • For each $k$, the total number of ways to choose $k$ elements from the first subset and $r-k$ elements from the second subset is the product $\binom{m}{k} \binom{n}{r-k}$.

5. Summing Over All Possible Values of $k$:

   • Since $k$ can range from 0 to $r$, we sum over all possible values of $k$ to count all possible ways to choose $r$ elements from $m+n$:

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

6. Conclusion:

   • This is exactly the expression on the right-hand side of the given identity. Hence, the identity is proven:

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

This completes the proof using a combinatorial argument.

Would you like to see more detailed steps, or do you have any specific questions?

Related Questions:

1. What is the binomial theorem, and how is it related to binomial coefficients?
2. How can this identity be used in probability theory or statistics?
3. Can this identity be proven using algebraic methods instead of combinatorial arguments?
4. What are other combinatorial identities involving binomial coefficients?
5. How does Vandermonde's identity extend to multinomial coefficients?

Tip:

When dealing with combinatorial identities, try interpreting them in terms of choosing elements from sets or subsets. This approach often provides an intuitive proof.