In a survey of the usage of three toothpaste A, B and C, it is found that 60 people like A,55 like B 40 like C, 20 like A and B, 15 like A and C and 10 like all three toothpastes. Find the following :

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.

In a survey of the usage of three toothpaste A, B and C, it is found that 60 people like A, 55 like B, 40 like C, 20 like A and B, 15 like A and C and 10 like all three toothpastes. Find the following:

This detailed solution explains how to analyze a survey of toothpaste preferences using set theory and Venn diagrams. We explore concepts such as the inclusion-exclusion principle to determine the number of people who like only one, two, or all three toothpaste brands A, B, and C. Suitable for Grades 10-12.

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