Question

A group of college students were asked about their TV watching habits. Of those surveyed, 28 students watch The Walking Dead, 19 watch The Blacklist, and 24 watch Game of Thrones. Additionally, 16 watch The Walking Dead and The Blacklist, 14 watch The Walking Dead and Game of Thrones, and 10 watch The Blacklist and Game of Thrones. There are 8 students who watch all three shows. How many students surveyed watched at least one of the shows?

Answer 1

Answer:

39 of the students surveyed watched at least one of the shows.

Step-by-step explanation:

We can solve this question building the Venn's Diagram of these sets.

I am going to say that:

A are those who watch The Walking Dead.

B are those who watch The Blacklist.

C are those who watch Game of Thrones.

We have that:

$A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)$

In which a are those who only watch The Walking Dead, $A \cap B$ are those who watch The Walking Dead and The Blacklist, but not Game of Thrones, $A \cap C$ are those who watch The Walking Dead and Game of Thrones, but not The Blacklist, and $A \cap B \cap C$ are those who watch all three of these shows.

By the same logic, we also have that:

$B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)$

$C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

How many students surveyed watched at least one of the shows?

This is

$T = a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

We have to find these values, starting from the intersection of A, B and C.

8 students who watch all three shows.

This means that

$A \cap B \cap C = 8$

10 watch The Blacklist and Game of Thrones.

This also includes those who watch all three shows. So

$(B \cap C) + (A \cap B \cap C) = 10$

$(B \cap C) = 2$

14 watch The Walking Dead and Game of Thrones

This also includes those who watch all three shows. So

$(A \cap C) + (A \cap B \cap C) = 14$

$(A \cap C) = 6$

16 watch The Walking Dead and The Blacklist

This also includes those who watch all three shows. So

$(A \cap B) + (A \cap B \cap C) = 16$

$(A \cap B) = 8$

24 watch Game of Thrones.

This means that $C = 24$

$C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

$24 = c + 6 + 2 + 8$

$c = 8$

19 watch The Blacklist

This means that $B = 19$

$B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)$

$19 = b + 8 + 2 + 8$

$b = 1$

28 students watch The Walking Dead

This means that $A = 28$

$A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)$

$28 = a + 8 + 6 + 8$

$a = 6$

Finally

$T = a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

$T = 6 + 1 + 8 + 8 + 6 + 2 + 8$

$T = 39$

39 of the students surveyed watched at least one of the shows.