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

Options

A. (a)`""^(n+1) C_(r-1) `
B. (b)` ""^(n+2) C_(r+1)`
C. (c)` ""^(n+2) C_(r)`
D. (d)` ""^(n+2)C_r`

Answer

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

### Step-by-Step Solution:

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

2. **Rearrange the expression**:
We can express the term \( 2\binom{n}{r-1} \) as \( \binom{n}{r-1} + \binom{n}{r-1} \):
\[ \binom{n}{r} + \binom{n}{r-1} + \binom{n}{r-1} + \binom{n}{r-2} \]

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

4. **Apply the property of binomial coefficients**:
Using the property \( \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \):
- For the first group:
\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]
- For the second group:
\[ \binom{n}{r-1} + \binom{n}{r-2} = \binom{n+1}{r-1} \]

5. **Combine the results**:
Now we have:
\[ \binom{n+1}{r} + \binom{n+1}{r-1} \]

6. **Apply the property again**:
Using the property again:
\[ \binom{n+1}{r} + \binom{n+1}{r-1} = \binom{n+2}{r} \]

7. **Final result**:
Therefore, the expression simplifies to:
\[ \binom{n+2}{r} \]

### Conclusion:
The value of the expression \( \binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2} \) is:
\[ \binom{n+2}{r} \]