IF `n=""^m C_2` then `"^n C_2` is equal to

Options

A. `"^(m+1)C_4`
B. `"^(m+2)C_4`
C. `"^(m-1)C_4`
D. `3. "^(m+1)C_4`

Answer

To solve the problem where \( n = \binom{m}{2} \), we need to find \( \binom{n}{2} \).

### Step-by-Step Solution:

1. **Understanding \( n \)**:
Given \( n = \binom{m}{2} \), we can express this as:
\[ n = \frac{m(m-1)}{2} \]

2. **Finding \( \binom{n}{2} \)**:
We need to calculate \( \binom{n}{2} \):
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]

3. **Substituting for \( n \)**:
Substitute \( n = \frac{m(m-1)}{2} \) into the equation:
\[ \binom{n}{2} = \frac{\left(\frac{m(m-1)}{2}\right)\left(\frac{m(m-1)}{2} - 1\right)}{2} \]

4. **Simplifying \( n - 1 \)**:
Calculate \( n - 1 \):
\[ n - 1 = \frac{m(m-1)}{2} - 1 = \frac{m(m-1) - 2}{2} \]

5. **Substituting \( n - 1 \)**:
Substitute \( n - 1 \) back into the equation:
\[ \binom{n}{2} = \frac{\left(\frac{m(m-1)}{2}\right)\left(\frac{m(m-1) - 2}{2}\right)}{2} \]

6. **Combining the terms**:
Now, we can simplify the expression:
\[ \binom{n}{2} = \frac{m(m-1)(m(m-1) - 2)}{8} \]

7. **Final Expression**:
The final expression for \( \binom{n}{2} \) can be represented as:
\[ \binom{n}{2} = \frac{m(m-1)(m^2 - m - 2)}{8} \]

### Conclusion:
Thus, the value of \( \binom{n}{2} \) is:
\[ \frac{m(m-1)(m^2 - m - 2)}{8} \]