In a market research survey 217 persons were asked if they – during the last year – had used any
of three competing products A, B and C.
97 persons had used A, and of these 43 had used A only.
89 persons had used B, and of these 10 had used B only.
137 persons had used C, and of these 39 had used C only.
62 persons have used B and C.
a) Are there any persons who have used all three products A, B and C?
b) How many persons have not used any of the products A, B and C?
c) How many persons have used products B or C?


To solve this problem, we need to analyze the given information and use set theory to find the answers to each question. Let's define:

- \( n(A) \) = number of persons who used product A
- \( n(B) \) = number of persons who used product B
- \( n(C) \) = number of persons who used product C
- \( n(A \cap B) \) = number of persons who used both products A and B
- \( n(A \cap C) \) = number of persons who used both products A and C
- \( n(B \cap C) \) = number of persons who used both products B and C
- \( n(A \cap B \cap C) \) = number of persons who used all three products A, B, and C

### Given Data:

- \( n(A) = 97 \)
- \( n(B) = 89 \)
- \( n(C) = 137 \)
- \( n(A \text{ only}) = 43 \)
- \( n(B \text{ only}) = 10 \)
- \( n(C \text{ only}) = 39 \)
- \( n(B \cap C) = 62 \)

### a) Are there any persons who have used all three products A, B, and C?

We can calculate \( n(A \cap B \cap C) \) by using the principle of inclusion-exclusion.

1. **Finding the number of people who used only A and B**:

\[
n(A \cap B \text{ only}) = n(A \cap B) - n(A \cap B \cap C)
\]

2. **Finding the number of people who used only A and C**:

\[
n(A \cap C \text{ only}) = n(A \cap C) - n(A \cap B \cap C)
\]

3. **Finding the number of people who used only B and C**:

\[
n(B \cap C \text{ only}) = n(B \cap C) - n(A \cap B \cap C)
\]

To compute \( n(A \cap B \cap C) \), we need to express all used combinations:

\[
n(A) = n(A \text{ only}) + n(A \cap B \text{ only}) + n(A \cap C \text{ only}) + n(A \cap B \cap C)
\]

\[
n(B) = n(B \text{ only}) + n(A \cap B \text{ only}) + n(B \cap C \text{ only}) + n(A \cap B \cap C)
\]

\[
n(C) = n(C \text{ only}) + n(A \cap C \text{ only}) + n(B \cap C \text{ only}) + n(A \cap B \cap C)
\]

Using the total number of persons surveyed, we have:

\[
n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)
\]

Substitute the known values:

\[
n(A \cup B \cup C) = 97 + 89 + 137 - n(A \cap B) - 62 - n(A \cap C) + n(A \cap B \cap C)
\]

We also know that:

\[
n(A \cup B \cup C) = 217 - x \quad \text{where } x \text{ is the number of people who have not used any product.}
\]

Using the information on the only users:

\[
97 - 43 = 54 \implies n(A \cap B) + n(A \cap C) - n(A \cap B \cap C) = 54
\]

\[
89 - 10 = 79 \implies n(A \cap B) + n(B \cap C) - n(A \cap B \cap C) = 79
\]

\[
137 - 39 = 98 \implies n(A \cap C) + n(B \cap C) - n(A \cap B \cap C) = 98
\]

Simultaneous solving we get:

\[
n(A \cap B \cap C) = 10
\]

### b) How many persons have not used any of the products A, B, and C?

We use:

\[
x = 217 - n(A \cup B \cup C) = 217 - (97 + 89 + 137 - 54 - 79 - 98 + 10) = 217 - 202 = 15
\]

Thus, 15 people haven't used any of the products.

### c) How many persons have used products B or C?

\[
n(B \cup C) = n(B) + n(C) - n(B \cap C) = 89 + 137 - 62 = 164
\]

### Final Answers:

a) Yes, 10 people have used all three products A, B, and C.

b) 15 persons have not used any of the products A, B, and C.

c) 164 persons have used products B or C.

Would you like more details or have any questions?

---

### Related Questions:

1. How many people used exactly two products?
2. What is the probability that a randomly selected person used only product A?
3. How many people used product A or B but not C?
4. How can the principle of inclusion-exclusion be applied to find the number of people who used two or more products?
5. If another product D was introduced, how would the survey data change?

### Tip:

For solving problems involving sets and surveys, the principle of inclusion-exclusion is very useful to account for overlapping sets.

In a market research survey 217 persons were asked if they – during the last year – had used any of three competing products A, B and C. 97 persons had used A, and of these 43 had used A only. 89 persons had used B, and of these 10 had used B only. 137 persons had used C, and of these 39 had used C only. 62 persons have used B and C. a) Are there any persons who have used all three products A, B and C? b) How many persons have not used any of the products A, B and C? c) How many persons have used products B or C?

Solve a market research problem where 217 participants were surveyed about their usage of products A, B, and C. Learn how to determine the number of people who used all three products, who didn’t use any, and who used either B or C using set theory and the inclusion-exclusion principle.

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