`""^(n)C_(r)+^(n)C_(r+1)=^(n+1)C_(x)`,then x=?

Options

A. r
B. r-1
C. n
D. r+1

Answer

To solve the problem \( \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{x} \), we will use the property of combinations.

### Step-by-Step Solution:

1. **Understand the Combination Property**: We know from the properties of combinations that:
\[ \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \]
This property states that the sum of two consecutive combinations from the same \( n \) is equal to the combination of \( n+1 \) taken \( r+1 \).

2. **Set Up the Equation**: Given the equation:
\[ \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{x} \]
We can replace the left-hand side using the property we just mentioned:
\[ \binom{n+1}{r+1} = \binom{n+1}{x} \]

3. **Compare the Indices**: Since the combinations are equal, we can equate the indices:
\[ x = r + 1 \]

4. **Conclusion**: Therefore, the value of \( x \) is:
\[ x = r + 1 \]

### Final Answer:
\[ x = r + 1 \]