If `mltn` then `mDn=mCn` else `mDn=nCm`
If `m=n` then `mEn=mn` else `mEn=mCn` where `mCn=(m)/(n)`
Find the value of `(12D60)E(2C10)`:

Options

A. 1
B. `(1)/(5)`
C. `1//25`
D. 0

Answer

To solve the problem, we need to evaluate the expression \( (12D60)E(2C10) \) using the definitions provided.

### Step-by-Step Solution:

1. **Evaluate \( 12D60 \)**:
- According to the definition, if \( m < n \), then \( mDn = mCn \).
- Here, \( m = 12 \) and \( n = 60 \). Since \( 12 < 60 \), we use the formula \( mDn = mCn \).
- Now, we need to calculate \( 12C60 \).
- The formula for \( mCn \) is given as \( \frac{m}{n} \).
- Therefore, \( 12C60 = \frac{12}{60} = \frac{1}{5} \).
- Thus, \( 12D60 = \frac{1}{5} \).

2. **Evaluate \( 2C10 \)**:
- Using the same formula \( mCn = \frac{m}{n} \), where \( m = 2 \) and \( n = 10 \).
- We calculate \( 2C10 = \frac{2}{10} = \frac{1}{5} \).

3. **Evaluate \( (12D60)E(2C10) \)**:
- We have found that \( 12D60 = \frac{1}{5} \) and \( 2C10 = \frac{1}{5} \).
- Now we need to evaluate \( \left( \frac{1}{5} \right) E \left( \frac{1}{5} \right) \).
- According to the definition, if \( m = n \), then \( mEn = mn \).
- Here, since both values are equal (\( \frac{1}{5} = \frac{1}{5} \)), we use the formula \( mEn = mn \).
- Therefore, \( \left( \frac{1}{5} \right) E \left( \frac{1}{5} \right) = \left( \frac{1}{5} \right) \times \left( \frac{1}{5} \right) = \frac{1}{25} \).

### Final Answer:
The value of \( (12D60)E(2C10) \) is \( \frac{1}{25} \).

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