`""^(n-2)C_(r)+2""^(n-2)C_(r-1)+""^(n-2)C_(r-2)` equals :

Options

A. `""^(n+1)C_(r)`
B. `""^(n)C_(r)`
C. `""^(n)C_(r+1)`
D. `""^(n-1)C_(r)`

Answer

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

### Step-by-Step Solution:

1. **Write down the expression**:
\[ \binom{n-2}{r} + 2 \binom{n-2}{r-1} + \binom{n-2}{r-2} \]

2. **Use the identity for binomial coefficients**:
We know that:
\[ \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} \]
We can apply this identity to the terms in our expression.

3. **Rearranging the expression**:
Notice that we can express \( \binom{n-2}{r} \) and \( \binom{n-2}{r-2} \) in terms of \( \binom{n-1}{r} \) and \( \binom{n-1}{r-1} \):
\[ \binom{n-2}{r} = \binom{n-1}{r} - \binom{n-1}{r-1} \]
\[ \binom{n-2}{r-1} = \binom{n-1}{r-1} - \binom{n-1}{r-2} \]
\[ \binom{n-2}{r-2} = \binom{n-1}{r-2} \]

4. **Substituting back into the expression**:
Substitute these identities back into the original expression:
\[ \left( \binom{n-1}{r} - \binom{n-1}{r-1} \right) + 2 \left( \binom{n-1}{r-1} - \binom{n-1}{r-2} \right) + \binom{n-1}{r-2} \]

5. **Combine like terms**:
Simplifying the expression:
\[ \binom{n-1}{r} - \binom{n-1}{r-1} + 2\binom{n-1}{r-1} - 2\binom{n-1}{r-2} + \binom{n-1}{r-2} \]
This simplifies to:
\[ \binom{n-1}{r} + \binom{n-1}{r-1} - \binom{n-1}{r-2} \]

6. **Use the identity again**:
Now, we can apply the binomial coefficient identity again:
\[ \binom{n-1}{r} + \binom{n-1}{r-1} = \binom{n}{r} \]

7. **Final Result**:
Therefore, the final result is:
\[ \binom{n}{r} \]

### Conclusion:
The expression \( \binom{n-2}{r} + 2 \binom{n-2}{r-1} + \binom{n-2}{r-2} \) equals \( \binom{n}{r} \).