To solve the problem, we need to determine how many days C alone can complete the work, given the work rates of A, B, and the combined work rate of A, B, and C.
### Step-by-Step Solution:
1. **Determine the work done by A and B individually:**
- A can complete the work in 12 days. Therefore, A's work rate is:
\[
\text{Work rate of A} = \frac{1}{12} \text{ (work per day)}
\]
- B can complete the work in 15 days. Therefore, B's work rate is:
\[
\text{Work rate of B} = \frac{1}{15} \text{ (work per day)}
\]
2. **Calculate the combined work rate of A and B:**
- The combined work rate of A and B is:
\[
\text{Combined work rate of A and B} = \frac{1}{12} + \frac{1}{15}
\]
- To add these fractions, find a common denominator (which is 60):
\[
\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}
\]
- Thus,
\[
\text{Combined work rate of A and B} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20}
\]
3. **Determine the combined work rate of A, B, and C:**
- A, B, and C together can finish the work in 4 days, so their combined work rate is:
\[
\text{Combined work rate of A, B, and C} = \frac{1}{4}
\]
4. **Set up the equation to find C's work rate:**
- We know:
\[
\text{Work rate of A + Work rate of B + Work rate of C} = \text{Work rate of A, B, and C}
\]
- Substituting the known values:
\[
\frac{3}{20} + \text{Work rate of C} = \frac{1}{4}
\]
5. **Convert \(\frac{1}{4}\) to a fraction with a denominator of 20:**
- \(\frac{1}{4} = \frac{5}{20}\)
6. **Solve for C's work rate:**
- Now, substituting back into the equation:
\[
\frac{3}{20} + \text{Work rate of C} = \frac{5}{20}
\]
- Rearranging gives:
\[
\text{Work rate of C} = \frac{5}{20} - \frac{3}{20} = \frac{2}{20} = \frac{1}{10}
\]
7. **Determine the number of days C alone can complete the work:**
- If C's work rate is \(\frac{1}{10}\), then C can complete the work in:
\[
\text{Days taken by C} = \frac{1}{\text{Work rate of C}} = \frac{1}{\frac{1}{10}} = 10 \text{ days}
\]
### Final Answer:
C alone can complete the work in **10 days**.
