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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!

C(n+1, r), C(n, r) and C(n, r-1) are binomial coefficients and satisfies the condition C(n+1, r) = C(n, r) + C(n, r-1), which one of the following exist?

Learn how to solve a binomial coefficient problem involving Pascal's identity. The problem asks for the correct relation between binomial terms C(n+1, r), C(n, r), and C(n, r-1). The solution involves analyzing multiple options to find the one that fits Pascal's Identity. Suitable for students in Grades 10-12.

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