Let `f(n)= sum_(k=1)^(n) k^2 ^"(n )C_k)^ 2` then the value of f(5) equals

Options

A. `1000`
B. `1250`
C. `1750`
D. `2500`

Explanation

To solve the problem, we need to evaluate the function \( f(n) = \sum_{k=1}^{n} k^2 \binom{n}{k}^2 \) for \( n = 5 \).

### Step-by-Step Solution:

1. **Substitute \( n = 5 \)**:
\[ f(5) = \sum_{k=1}^{5} k^2 \binom{5}{k}^2 \]

2. **Expand the summation**:
\[ f(5) = 1^2 \binom{5}{1}^2 + 2^2 \binom{5}{2}^2 + 3^2 \binom{5}{3}^2 + 4^2 \binom{5}{4}^2 + 5^2 \binom{5}{5}^2 \]

3. **Calculate each term**:
- For \( k = 1 \):
\[ 1^2 \binom{5}{1}^2 = 1 \cdot 5^2 = 25 \]
- For \( k = 2 \):
\[ 2^2 \binom{5}{2}^2 = 4 \cdot \left(\frac{5 \cdot 4}{2 \cdot 1}\right)^2 = 4 \cdot 10^2 = 400 \]
- For \( k = 3 \):
\[ 3^2 \binom{5}{3}^2 = 9 \cdot \left(\frac{5 \cdot 4 \cdot 3}{3 \cdot 2 \cdot 1}\right)^2 = 9 \cdot 10^2 = 900 \]
- For \( k = 4 \):
\[ 4^2 \binom{5}{4}^2 = 16 \cdot 5^2 = 16 \cdot 25 = 400 \]
- For \( k = 5 \):
\[ 5^2 \binom{5}{5}^2 = 25 \cdot 1^2 = 25 \]

4. **Sum all the terms**:
\[ f(5) = 25 + 400 + 900 + 400 + 25 \]
\[ f(5) = 25 + 400 = 425 \]
\[ f(5) = 425 + 900 = 1325 \]
\[ f(5) = 1325 + 400 = 1725 \]
\[ f(5) = 1725 + 25 = 1750 \]

5. **Final Result**:
\[ f(5) = 1750 \]

### Summary:
The value of \( f(5) \) is \( 1750 \).