For `2<=r<=n,((n),(r))+2((n),(r-1))+((n),(r-2))` is equal to

Options

A. `((n+1),(r-1))`
B. `2((n+1),(r+1))`
C. `((n+2),(r))`
D. `2((n+2),(r))`

Explanation

To solve the problem, we need to simplify the expression given:

\[ \binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2} \]

### Step 1: Rewrite the expression using binomial coefficients
The expression can be rewritten in terms of binomial coefficients:

\[ \binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2} \]

### Step 2: Apply the property of binomial coefficients
We can use the property of binomial coefficients which states:

\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]

Using this property, we can combine the first two terms:

\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]

Thus, we can rewrite our expression as:

\[ \binom{n+1}{r} + \binom{n}{r-1} \]

### Step 3: Apply the property again
Now we can apply the property again to the new expression:

\[ \binom{n+1}{r} + \binom{n}{r-1} = \binom{n+1}{r} + \binom{n+1}{r-1} \]

According to the property, this can be simplified to:

\[ \binom{n+2}{r} \]

### Final Result
Thus, we find that:

\[ \binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2} = \binom{n+2}{r} \]

### Conclusion
The final answer is:

\[ \binom{n+2}{r} \]