`""^(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 \( \binom{n-2}{r} + 2 \cdot \binom{n-2}{r-1} + \binom{n-2}{r-2} \), we can use the properties of binomial coefficients.

### Step-by-Step Solution:

1. **Identify the Terms**: We have three terms in the expression:
- \( \binom{n-2}{r} \)
- \( 2 \cdot \binom{n-2}{r-1} \)
- \( \binom{n-2}{r-2} \)

2. **Use the Binomial Coefficient Identity**: Recall the identity:
\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]
We can manipulate our expression to fit this identity.

3. **Rewrite the Expression**: We can rewrite the expression as:
\[ \binom{n-2}{r} + \binom{n-2}{r-1} + \binom{n-2}{r-1} + \binom{n-2}{r-2} \]
This allows us to group terms.

4. **Group the Terms**: Now, group the terms:
\[ \left( \binom{n-2}{r} + \binom{n-2}{r-1} \right) + \left( \binom{n-2}{r-1} + \binom{n-2}{r-2} \right) \]

5. **Apply the Identity**: Using the identity:
- For the first group:
\[ \binom{n-2}{r} + \binom{n-2}{r-1} = \binom{n-1}{r} \]
- For the second group:
\[ \binom{n-2}{r-1} + \binom{n-2}{r-2} = \binom{n-1}{r-1} \]

6. **Combine the Results**: Now we can combine the results:
\[ \binom{n-1}{r} + \binom{n-1}{r-1} \]

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

### Conclusion:
Thus, the original expression simplifies to:
\[ \binom{n}{r} \]