The problem you shared involves binomial coefficients and asks to identify the correct relationship that holds between different binomial terms.

Given:

• $C(n+1, r) = C(n, r) + C(n, r-1)$ is the key identity based on Pascal's identity for binomial coefficients.

We need to identify which of the provided options follows this identity. Let's analyze them:

Option A:

$C(n-1, r) + C(n-1, r-1) = C(n, r-1)$ This is Pascal's identity but shifted for the lower terms $n-1$. This looks plausible, but let's check other options.

Option B:

$C(n-1, r) + C(n-1, r+1) = C(n, r)$ This does not align with Pascal’s identity as the second term involves $r+1$.

Option C:

$C(n-1, r-1) + C(n-1, r-2) = C(n, r-1)$ This does not fit either because Pascal’s identity involves only $r$ and $r-1$, but here we have $r-1$ and $r-2$.

Option D:

$C(n-1, r) + C(n-1, r-2) = C(n, r-1)$ This again doesn’t follow Pascal’s identity.

Conclusion:

Option A is the correct one. It follows Pascal's identity: $C(n, r) = C(n-1, r) + C(n-1, r-1)$

Would you like more details on how Pascal's identity works? Let me know if you have any other questions!

Related questions:

1. What is Pascal’s triangle, and how does it relate to binomial coefficients?
2. How are binomial coefficients used in the expansion of $(x + y)^n$?
3. Can Pascal’s identity be derived using combinatorial arguments?
4. How do binomial coefficients relate to combinations and counting problems?
5. How does Pascal’s identity apply in recursive algorithms?

Tip:

When dealing with binomial coefficients, remembering Pascal's triangle and identity will help you quickly verify relationships!