1. **State the problem:** We need to find the minimum total price for a school trip for 100 students to a museum where each group can have a maximum of 25 people including at least one teacher. 2. **Understand the constraints:** Each group must have at least one teacher and the total number of people per group (students + teachers) cannot exceed 25. 3. **Determine the number of groups:** Since there are 100 students and each group can have at most 25 people, the minimum number of groups is $$\lceil \frac{100}{25 - 1} \rceil$$ because each group must have at least one teacher, so maximum students per group is 24. 4. Calculate the number of groups: $$\frac{100}{24} = 4.166\ldots$$ So, the minimum number of groups is 5. 5. **Calculate total number of teachers:** Since each group must have at least one teacher, total teachers = 5. 6. **Calculate total number of people:** Total people = 100 students + 5 teachers = 105. 7. **Calculate total entrance fee:** Entrance fee per person is $a$, so total entrance fee = $$a \times 105$$. 8. **Calculate total guide cost:** Guide cost per group is $b$, so total guide cost = $$b \times 5$$. 9. **Calculate total minimum price:** Total price = entrance fees + guide costs = $$105a + 5b$$. **Final answer:** $$\boxed{105a + 5b}$$