If `n = ""^(k)C_2`, the value of `""^(n)C_2` is:

Options

A. a. `2 (""^(K+2)C_4)`
B. b. `(n-2)!`
C. c. `(k-2)!`
D. d. `3 (""^(K+3) C_4)`

Answer

To solve the problem, we need to find the value of \( \binom{n}{2} \) given that \( n = \binom{k}{2} \).

### Step-by-Step Solution:

1. **Understanding the Given Expression**:
We know that \( n = \binom{k}{2} \). The formula for combinations is given by:
\[ \binom{k}{2} = \frac{k!}{2!(k-2)!} = \frac{k(k-1)}{2} \]
Therefore, we can express \( n \) as:
\[ n = \frac{k(k-1)}{2} \]

2. **Finding \( \binom{n}{2} \)**:
Now we need to calculate \( \binom{n}{2} \):
\[ \binom{n}{2} = \frac{n!}{2!(n-2)!} \]
Substituting \( n = \frac{k(k-1)}{2} \):
\[ \binom{n}{2} = \frac{\left(\frac{k(k-1)}{2}\right)!}{2!\left(\frac{k(k-1)}{2} - 2\right)!} \]

3. **Simplifying \( n - 2 \)**:
We need to find \( n - 2 \):
\[ n - 2 = \frac{k(k-1)}{2} - 2 = \frac{k(k-1) - 4}{2} \]

4. **Substituting Back**:
Now substituting \( n \) and \( n - 2 \) into the combination formula:
\[ \binom{n}{2} = \frac{\left(\frac{k(k-1)}{2}\right)!}{2! \left(\frac{k(k-1) - 4}{2}\right)!} \]

5. **Calculating \( \binom{n}{2} \)**:
The next step is to simplify this expression. We can express \( \binom{n}{2} \) in terms of \( k \):
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
Substituting \( n = \frac{k(k-1)}{2} \):
\[ \binom{n}{2} = \frac{\frac{k(k-1)}{2} \left(\frac{k(k-1)}{2} - 1\right)}{2} \]

6. **Final Simplification**:
Continuing to simplify:
\[ = \frac{\frac{k(k-1)}{2} \cdot \left(\frac{k(k-1) - 2}{2}\right)}{2} = \frac{k(k-1)(k(k-1) - 2)}{8} \]

7. **Conclusion**:
The final expression for \( \binom{n}{2} \) is:
\[ \binom{n}{2} = \frac{k(k-1)(k^2 - 3k + 2)}{8} \]