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

Options

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

Answer

To solve the expression \( \binom{n}{r+1} + 2 \binom{n}{r} + \binom{n}{r-1} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \binom{n}{r+1} + 2 \binom{n}{r} + \binom{n}{r-1} \]

### Step 2: Break down the terms
Notice that \( 2 \binom{n}{r} \) can be expressed as:
\[ \binom{n}{r} + \binom{n}{r} \]
Thus, we can rewrite the expression as:
\[ \binom{n}{r+1} + \binom{n}{r} + \binom{n}{r} + \binom{n}{r-1} \]

### Step 3: Group the terms
Now, we can group the terms:
\[ \binom{n}{r+1} + \binom{n}{r-1} + \binom{n}{r} + \binom{n}{r} \]

### Step 4: Apply the Hockey Stick Identity
According to the Hockey Stick Identity in combinatorics, we have:
\[ \binom{n}{r-1} + \binom{n}{r} = \binom{n+1}{r} \]
and
\[ \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \]

### Step 5: Combine the results
Using the Hockey Stick Identity:
\[ \binom{n}{r+1} + \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r+1} + \binom{n}{r} \]
Now, we can apply the Hockey Stick Identity again:
\[ \binom{n+1}{r+1} + \binom{n}{r} = \binom{n+2}{r+1} \]

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