If 1<=r<= n Then prove that c(n,r) + c(n,r+1) = c(n+1,r+1)

We are asked to prove that:

\[
C(n, r) + C(n, r+1) = C(n+1, r+1)
\]

where \(C(n, r)\) is the binomial coefficient, commonly denoted as "n choose r", which is defined as:

\[
C(n, r) = \frac{n!}{r!(n - r)!}
\]

### Proof:

Using the definition of the binomial coefficient:

1. **Left-hand side (LHS)**:
\[
C(n, r) + C(n, r+1)
\]

Substitute the definitions of \(C(n, r)\) and \(C(n, r+1)\):

\[
LHS = \frac{n!}{r!(n - r)!} + \frac{n!}{(r+1)!(n - r - 1)!}
\]

We now need to simplify this expression. First, let's factor out the common terms:

\[
LHS = \frac{n!}{r!(n - r)!} \left( 1 + \frac{r!(n - r)!}{(r+1)!(n - r - 1)!} \right)
\]

Simplify the expression inside the parentheses:

\[
1 + \frac{r!(n - r)!}{(r+1)!(n - r - 1)!} = 1 + \frac{1}{(r+1)(n - r)}
\]

Thus, the LHS becomes:

\[
LHS = \frac{n!}{r!(n - r)!} \cdot \frac{(r+1)(n - r) + 1}{(r+1)(n - r)}
\]

Now, simplify the numerator:

\[
(r+1)(n - r) + 1 = (r+1)n - (r+1)r + 1 = n + r + 1
\]

So we have:

\[
LHS = \frac{n!}{r!(n - r)!} \cdot \frac{n+1}{(r+1)(n - r)}
\]

Now, notice that:

\[
\frac{n!}{r!(n - r)!} \cdot \frac{1}{n - r} = \frac{n!}{r!(n - r + 1)!} = \frac{(n+1)!}{(r+1)!(n - r)!}
\]

Thus, we get:

\[
LHS = \frac{(n+1)!}{(r+1)!(n - r)!}
\]

This is precisely the binomial coefficient \(C(n+1, r+1)\), which is the **right-hand side (RHS)**.

### Conclusion:

\[
C(n, r) + C(n, r+1) = C(n+1, r+1)
\]

Thus, the identity is proven.

---

Do you have any questions or would you like more details?

Here are some related questions for further exploration:

1. What is the combinatorial interpretation of this identity?
2. How does Pascal’s Triangle relate to this binomial identity?
3. How would this proof change if we used induction instead of direct simplification?
4. Can this identity be generalized for more than two binomial coefficients?
5. How can you use this identity in a counting problem?

**Tip:** Binomial coefficients frequently appear in combinatorics, especially when counting subsets of a given set.

If 1 <= r <= n, then prove that C(n, r) + C(n, r+1) = C(n+1, r+1).

Explore a detailed proof of the binomial identity C(n, r) + C(n, r+1) = C(n+1, r+1), where C(n, r) represents the binomial coefficient. This step-by-step explanation includes key combinatorial concepts and simplifications, suitable for advanced high school or introductory college-level students.

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