`f(n)=sum_(r=1)^(n) [r^(2)(""^(n)C_(r)-""^(n)C_(r-1))+(2r+1)(""^(n)C_(r ))]`, then

Options

A. `f(30)=960`
B. `f(21)=483`
C. `f(16)=64`
D. `f(11)=44`

Explanation

To solve the given problem, we need to analyze the function \( f(n) \) defined as:

\[ f(n) = \sum_{r=1}^{n} \left[ r^2 \left( \binom{n}{r} - \binom{n}{r-1} \right) + (2r + 1) \binom{n}{r} \right] \]

### Step 1: Simplifying the Summation

We can rewrite the function as:

\[ f(n) = \sum_{r=1}^{n} \left[ r^2 \binom{n}{r} - r^2 \binom{n}{r-1} + (2r + 1) \binom{n}{r} \right] \]

This can be separated into two summations:

\[ f(n) = \sum_{r=1}^{n} r^2 \binom{n}{r} + \sum_{r=1}^{n} (2r + 1) \binom{n}{r} - \sum_{r=1}^{n} r^2 \binom{n}{r-1} \]

### Step 2: Evaluating Each Summation

The first term \( \sum_{r=1}^{n} r^2 \binom{n}{r} \) can be evaluated using the identity:

\[ \sum_{r=0}^{n} r^2 \binom{n}{r} = n(n-1)2^{n-2} + n2^{n-1} \]

The second term \( \sum_{r=1}^{n} (2r + 1) \binom{n}{r} \) can be simplified as follows:

\[ \sum_{r=1}^{n} (2r + 1) \binom{n}{r} = 2 \sum_{r=1}^{n} r \binom{n}{r} + \sum_{r=1}^{n} \binom{n}{r} \]

Using the identities:

\[ \sum_{r=0}^{n} r \binom{n}{r} = n2^{n-1} \]
\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n \]

We can substitute these values into our expression.

### Step 3: Combining the Results

After substituting the results into the expression for \( f(n) \), we can combine the terms. The third summation \( \sum_{r=1}^{n} r^2 \binom{n}{r-1} \) can also be evaluated similarly, leading to a cancellation of terms.

### Step 4: Final Expression

After simplification, we arrive at:

\[ f(n) = (n + 1)^2 - 1 \]

Thus, we can conclude that:

\[ f(n) = n^2 + 2n \]

### Step 5: Verification

To verify, we can substitute specific values of \( n \):

- For \( n = 30 \):
\[ f(30) = 31^2 - 1 = 961 - 1 = 960 \]

- For \( n = 21 \):
\[ f(21) = 22^2 - 1 = 484 - 1 = 483 \]

- For \( n = 16 \):
\[ f(16) = 17^2 - 1 = 289 - 1 = 288 \]

- For \( n = 11 \):
\[ f(11) = 12^2 - 1 = 144 - 1 = 143 \]

### Conclusion

The final result is:

\[ f(n) = n^2 + 2n \]