If $n=^{m}C_{2}$ then ${}^{n}C_{2}=$

Question

Accepted Answer

To solve the problem, we need to understand the relationship between the given combinations and how they lead to the correct answer, which is Option 4.

### Explanation:

1. **Understanding the Given Information:**
   - We are given that $ n = {}^{m}C_{2} $.
   - We need to find the value of $ {}^{n}C_{2} $.

2. **Expression for Combinations:**
   - The combination formula is given by:
     $$
     {}^{k}C_{r} = \frac{k!}{r!(k-r)!}
     $$
   - Specifically, for $ {}^{m}C_{2} $:
     $$
     {}^{m}C_{2} = \frac{m(m-1)}{2}
     $$
   - Therefore, $ n = \frac{m(m-1)}{2} $.

3. **Finding $ {}^{n}C_{2} $:**
   - We need to calculate $ {}^{n}C_{2} $ where $ n = \frac{m(m-1)}{2} $.
   - Using the combination formula:
     $$
     {}^{n}C_{2} = \frac{n(n-1)}{2}
     $$
   - Substitute $ n = \frac{m(m-1)}{2} $:
     $$
     {}^{n}C_{2} = \frac{\left(\frac{m(m-1)}{2}\right)\left(\frac{m(m-1)}{2} - 1\right)}{2}
     $$

4. **Simplifying the Expression:**
   - Simplifying the expression inside the combination:
     $$
     \frac{m(m-1)}{2} - 1 = \frac{m(m-1) - 2}{2}
     $$
   - Therefore:
     $$
     {}^{n}C_{2} = \frac{\left(\frac{m(m-1)}{2}\right)\left(\frac{m(m-1) - 2}{2}\right)}{2}
     $$

5. **Relating to the Options:**
   - We need to relate this expression to the options given.
   - Option 4 is $ 3({}^{m+1}C_{4}) $.
   - Calculate $ {}^{m+1}C_{4} $:
     $$
     {}^{m+1}C_{4} = \frac{(m+1)m(m-1)(m-2)}{24}
     $$
   - Therefore, $ 3({}^{m+1}C_{4}) $ becomes:
     $$
     3 \times \frac{(m+1)m(m-1)(m-2)}{24} = \frac{(m+1)m(m-1)(m-2)}{8}
     $$

6. **Verification:**
   - The expression for $ {}^{n}C_{2} $ simplifies to match $ \frac{(m+1)m(m-1)(m-2)}{8} $, confirming that Option 4 is indeed correct.

### Why Other Options are Incorrect:

- **Option 1: $ {}^{m+1}C_{4} $**
  - This option does not account for the factor of 3 needed to match the expression derived for $ {}^{n}C_{2} $.

- **Option 2: $ {}^{m-1}C_{4} $**
  - This option incorrectly uses $ m-1 $ instead of $ m+1 $, leading to a different and incorrect expression.

- **Option 3: $ {}^{m+2}C_{4} $**
  - This option incorrectly uses $ m+2 $, which does not align with the derived expression for $ {}^{n}C_{2} $.

In conclusion, Option 4 is the correct answer because it accurately represents the expression derived for $ {}^{n}C_{2} $ when $ n = {}^{m}C_{2} $.