In a city with a population of 1 million, there are three newspapers: Newspaper A, Newspaper B, and Newspaper C. A survey was conducted with 500 residents and it was found out that 300 read Newspaper A, 200 read Newspaper B, and 150 read Newspaper C. How many residents read all three newspapers?

1 Answers
Answered by suresh

To find out how many residents read all three newspapers, we can use the principle of inclusion and exclusion.

Let's define the following variables:
- $n(A)$: Number of residents who read Newspaper A
- $n(B)$: Number of residents who read Newspaper B
- $n(C)$: Number of residents who read Newspaper C
- $n(A cap B)$: Number of residents who read both Newspaper A and Newspaper B
- $n(A cap C)$: Number of residents who read both Newspaper A and Newspaper C
- $n(B cap C)$: Number of residents who read both Newspaper B and Newspaper C
- $n(A cap B cap C)$: Number of residents who read all three newspapers

According to the principle of inclusion and exclusion:
$$n(A cup B cup C) = n(A) + n(B) + n(C) - n(A cap B) - n(A cap C) - n(B cap C) + n(A cap B cap C)$$

Given the information in the question:
- $n(A) = 300$
- $n(B) = 200$
- $n(C) = 150$
- $n(A cap B) = n(A cap C) = n(B cap C) = 0$ (since there are no overlaps mentioned between any two newspapers)

Substitute the values into the formula:
$$500 = 300 + 200 + 150 - 0 - 0 - 0 + n(A cap B cap C)$$

Solving for $n(A cap B cap C)$:
$$n(A cap B cap C) = 500 - 300 - 200 - 150 = 50$$

Therefore, there are 50 residents who read all three newspapers - A, B, and C.

Answer for Question: In a city with a population of 1 million, there are three newspapers: Newspaper A, Newspaper B, and Newspaper C. A survey was conducted with 500 residents and it was found out that 300 read Newspaper A, 200 read Newspaper B, and 150 read Newspaper C. How many residents read all three newspapers?