Explanation: To solve this problem, we need to understand the concept of work and time, specifically how to calculate the combined work rate and individual work rates.
1. **Efficiency Calculation:**
- Amul can complete the work in 5 days, so his efficiency is \( \frac{1}{5} \) of the work per day.
- Parle can complete the work in 4 days, so his efficiency is \( \frac{1}{4} \) of the work per day.
- Let Goodday's efficiency be \( \frac{1}{x} \) of the work per day, where \( x \) is the number of days Goodday takes to complete the work alone.
2. **Combined Work Rate:**
- When Amul, Parle, and Goodday work together, their combined efficiency is the sum of their individual efficiencies.
- The combined efficiency is \( \frac{1}{5} + \frac{1}{4} + \frac{1}{x} \).
3. **Given Condition:**
- They complete the work in 2 days together, so their combined efficiency is \( \frac{1}{2} \) of the work per day.
4. **Setting Up the Equation:**
- We set up the equation based on the combined efficiency:
\[
\frac{1}{5} + \frac{1}{4} + \frac{1}{x} = \frac{1}{2}
\]
5. **Solving the Equation:**
- First, find a common denominator for the fractions on the left side:
\[
\frac{4}{20} + \frac{5}{20} + \frac{1}{x} = \frac{1}{2}
\]
- Combine the fractions:
\[
\frac{9}{20} + \frac{1}{x} = \frac{1}{2}
\]
- Isolate \( \frac{1}{x} \):
\[
\frac{1}{x} = \frac{1}{2} - \frac{9}{20}
\]
- Convert \( \frac{1}{2} \) to a fraction with a denominator of 20:
\[
\frac{1}{x} = \frac{10}{20} - \frac{9}{20} = \frac{1}{20}
\]
- Therefore, \( x = 20 \).
Thus, Goodday alone can complete the work in 20 days. This matches the correct answer, Option A.