Respuesta :
Answer:
6 students did not watch any one of these three movies.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the students that watched Part I.
-The set B represents the students that watched Part II.
-The set C represents the students that watched Part III.
-d is the number of students that did not watch any of these three movies.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of students that only watched Part I, [tex]A \cap B[/tex] is the number of students that watched both Part I and Part II, [tex]A \cap C[/tex] is the number of students that watched both Part I and Part III. And [tex]A \cap B \cap C[/tex] is the number of students that like all three parts.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There were 98 students suveyed. This means that:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 98[/tex]
We start finding the values from the intersection of three sets.
43 had watched all three parts. This means that [tex](A \cap B \cap C) = 43[/tex].
45 had watched both Parts II and III. This means that:
[tex](B \cap C) + (A \cap B \cap C) = 45[/tex]
[tex](B \cap C) = 2[/tex]
51 had watched both Parts I and III
[tex](A \cap C) + (A \cap B \cap C) = 51[/tex]
[tex](A \cap C) = 8[/tex]
52 had watched both Parts I and II
[tex](A \cap B) + (A \cap B \cap C) = 52[/tex]
[tex](A \cap B) = 9[/tex]
66 had watched Part III
[tex]C = 66[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]c + 8 + 2 + 43 = 66[/tex]
[tex]c = 13[/tex]
57 had watched Part II
[tex]B = 57[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]b + 2 + 9 + 43 = 57[/tex]
[tex]b = 3[/tex]
74 had watched Part I
[tex]A = 74[/tex]
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]a + 9 + 8 + 43 = 74[/tex]
[tex]a = 14[/tex]
How many students did not watch any one of these three movies?
We have to find d.
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 98[/tex]
[tex]14 + 3 + 13 + d + 9 + 8 + 2 + 43 = 98[/tex]
[tex]d = 98 - 92[/tex]
[tex]d = 6[/tex]
6 students did not watch any one of these three movies.
