Respuesta :
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:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a are those who only watch The Walking Dead, [tex]A \cap B[/tex] are those who watch The Walking Dead and The Blacklist, but not Game of Thrones, [tex]A \cap C[/tex] are those who watch The Walking Dead and Game of Thrones, but not The Blacklist, and [tex]A \cap B \cap C[/tex] are those who watch all three of these shows.
By the same logic, we also have that:
[tex]B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
How many students surveyed watched at least one of the shows?
This is
[tex]T = a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
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
[tex]A \cap B \cap C = 8[/tex]
10 watch The Blacklist and Game of Thrones.
This also includes those who watch all three shows. So
[tex](B \cap C) + (A \cap B \cap C) = 10[/tex]
[tex](B \cap C) = 2[/tex]
14 watch The Walking Dead and Game of Thrones
This also includes those who watch all three shows. So
[tex](A \cap C) + (A \cap B \cap C) = 14[/tex]
[tex](A \cap C) = 6[/tex]
16 watch The Walking Dead and The Blacklist
This also includes those who watch all three shows. So
[tex](A \cap B) + (A \cap B \cap C) = 16[/tex]
[tex](A \cap B) = 8[/tex]
24 watch Game of Thrones.
This means that [tex]C = 24[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]24 = c + 6 + 2 + 8[/tex]
[tex]c = 8[/tex]
19 watch The Blacklist
This means that [tex]B = 19[/tex]
[tex]B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]19 = b + 8 + 2 + 8[/tex]
[tex]b = 1[/tex]
28 students watch The Walking Dead
This means that [tex]A = 28[/tex]
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]28 = a + 8 + 6 + 8[/tex]
[tex]a = 6[/tex]
Finally
[tex]T = a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]T = 6 + 1 + 8 + 8 + 6 + 2 + 8[/tex]
[tex]T = 39[/tex]
39 of the students surveyed watched at least one of the shows.
