IF `""^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 equation \( \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{x} \), we can use the properties of binomial coefficients.

### Step-by-Step Solution:

1. **Understand the Binomial Coefficient Identity**:
We start with the identity:
\[ \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \]
This identity states that the sum of two consecutive binomial coefficients from the same row (n) is equal to a binomial coefficient from the next row (n+1) with a specific index.

2. **Apply the Identity**:
From the equation given in the problem:
\[ \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{x} \]
We can substitute the left-hand side using the identity:
\[ \binom{n+1}{r+1} = \binom{n+1}{x} \]

3. **Set the Indices Equal**:
Since the binomial coefficients are equal, we can equate their indices:
\[ r + 1 = x \]

4. **Solve for x**:
Thus, we find:
\[ x = r + 1 \]

### Final Answer:
The value of \( x \) is \( r + 1 \).