Let the three sets represent people who like toothpaste A, B, and C, respectively:

• $n(A) = 60$: Number of people who like A.
• $n(B) = 55$: Number of people who like B.
• $n(C) = 40$: Number of people who like C.
• $n(A \cap B) = 20$: Number of people who like both A and B.
• $n(A \cap C) = 15$: Number of people who like both A and C.
• $n(B \cap C) = 10$: Number of people who like all three, i.e., $n(A \cap B \cap C) = 10$.

We need to find the following:

1. The number of people who like A only.

This is given by: $n(A \text{ only}) = n(A) - [n(A \cap B) + n(A \cap C) - n(A \cap B \cap C)]$ $n(A \text{ only}) = 60 - [20 + 15 - 10] = 60 - 25 = 35$

2. The number of people who like B only.

Similarly: $n(B \text{ only}) = n(B) - [n(A \cap B) + n(B \cap C) - n(A \cap B \cap C)]$ $n(B \text{ only}) = 55 - [20 + 10 - 10] = 55 - 20 = 35$

3. The number of people who like C only.

$n(C \text{ only}) = n(C) - [n(A \cap C) + n(B \cap C) - n(A \cap B \cap C)]$ $n(C \text{ only}) = 40 - [15 + 10 - 10] = 40 - 15 = 25$

4. The number of people who like both A and B but not C.

This is: $n(A \cap B \text{ but not } C) = n(A \cap B) - n(A \cap B \cap C)$ $n(A \cap B \text{ but not } C) = 20 - 10 = 10$

5. The number of people who like all three toothpastes.

This is already given: $n(A \cap B \cap C) = 10$

Summary of findings:

1. People who like A only: 35
2. People who like B only: 35
3. People who like C only: 25
4. People who like both A and B but not C: 10
5. People who like all three toothpastes: 10

Would you like more details or have any further questions?

Here are some relative questions for you to consider:

1. How many people like at least one toothpaste?
2. How many people like both A and C but not B?
3. What is the number of people who like exactly two toothpastes?
4. How many people like either A or B but not C?
5. How many people like exactly one toothpaste?

Tip: When working with sets and surveys, drawing a Venn diagram can help you visually organize the data and simplify the calculations.