`""^n C_(r+1) +""^n C_(r-1)+2 "^n C_r=`

Options

A. `"^(n+1)C_r`
B. `"^(n+1)C_(r+1)`
C. `"^(n+2)C_(r+1)`
D. none of these

Answer

To solve the expression \( \binom{n}{r+1} + \binom{n}{r-1} + 2 \binom{n}{r} \), we can use the properties of binomial coefficients.

### Step-by-Step Solution:

1. **Understanding Binomial Coefficients**:
We start with the expression \( \binom{n}{r+1} + \binom{n}{r-1} + 2 \binom{n}{r} \). The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose \( k \) elements from a set of \( n \) elements.

2. **Using the Identity**:
We can use the identity:
\[ \binom{n}{r+1} + \binom{n}{r} = \binom{n+1}{r+1} \]
This identity states that the sum of two consecutive binomial coefficients equals the binomial coefficient of the next higher row.

3. **Rearranging the Terms**:
To apply the identity, we can rearrange our expression:
\[ \binom{n}{r+1} + \binom{n}{r-1} + 2 \binom{n}{r} = \binom{n}{r+1} + \binom{n}{r} + \binom{n}{r} + \binom{n}{r-1} \]
This allows us to group the terms \( \binom{n}{r+1} + \binom{n}{r} \) and \( \binom{n}{r} + \binom{n}{r-1} \).

4. **Applying the Identity**:
Now we apply the identity:
\[ \binom{n}{r+1} + \binom{n}{r} = \binom{n+1}{r+1} \]
and
\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]
Thus, we can rewrite our expression as:
\[ \binom{n+1}{r+1} + \binom{n+1}{r} \]

5. **Final Application of the Identity**:
Finally, we apply the identity again:
\[ \binom{n+1}{r+1} + \binom{n+1}{r} = \binom{n+2}{r+1} \]

### Conclusion:
Thus, we conclude that:
\[ \binom{n}{r+1} + \binom{n}{r-1} + 2 \binom{n}{r} = \binom{n+2}{r+1} \]