`""^(n)C_(r)+2""^(n)C_(r-1)+""^(n)C_(r-2)` is equal to

Options

A. `""^(n+1)C_(r)`
B. `""^(n+1)C_(r+1)`
C. `""^(n+2)C_(r)`
D. `""^(n+2)C_(r+1)`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Binomial Coefficient**:
Recall that \( \binom{n}{r} \) represents the number of ways to choose \( r \) elements from a set of \( n \) elements.

2. **Rearranging the Expression**:
The expression can be rewritten as:
\[ \binom{n}{r} + \binom{n}{r-1} + \binom{n}{r-1} + \binom{n}{r-2} \]
This shows that we have two instances of \( \binom{n}{r-1} \).

3. **Combining Like Terms**:
Combine the terms:
\[ \binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2} \]

4. **Using the Hockey Stick Identity**:
According to the hockey stick identity in combinatorics, we have:
\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]
Therefore, we can apply this identity to our expression:
\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]
Thus, we can rewrite the expression as:
\[ \binom{n+1}{r} + \binom{n}{r-1} \]

5. **Applying the Hockey Stick Identity Again**:
Now, we can apply the hockey stick identity again:
\[ \binom{n+1}{r} + \binom{n}{r-1} = \binom{n+1}{r} + \binom{n+1}{r-1} = \binom{n+2}{r} \]

6. **Final Result**:
Therefore, the final result is:
\[ \binom{n+2}{r} \]

### Conclusion:
Thus, the expression \( \binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2} \) simplifies to \( \binom{n+2}{r} \).