Respuesta :
Answer:
The total number of family members is 21.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of these sets.
I am going to say that:
-The set A represents those that would not go to a park.
-The set B represents those who would not go to a beach.
-The set C represents those who would not go to the family cottage.
The value d represents those who would go to all three places.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a are those that would only not go to a park, [tex]A \cap B[/tex] are those who would not got to a park or to the beach, [tex]A \cap C[/tex] are those who would not go to a park or to the famili cottage. And [tex]A \cap B \cap C[/tex] are those that would not go to any of these places.
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 values:
[tex]a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
The total number of family members is the sum of all these values:
[tex]T = a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
We start finding the values from the intersection of the three sets
5 would not go to a park or a beach or to the family cottage.
This means that [tex]A \cap B \cap C = 5[/tex]
1 would go to all three places. This means that [tex]d = 1[/tex].
8 would go to neither a park nor the family cottage
This means that:
[tex]A \cap C + (A \cap B \cap C) = 8[/tex]
[tex]A \cap C = 3[/tex]
8 would go to neither a beach nor the family cottage
[tex]B \cap C + (A \cap B \cap C) = 8[/tex]
[tex]B \cap C = 3[/tex]
7 would go to neither a park nor a beach
[tex]A \cap B + (A \cap B \cap C) = 7[/tex]
[tex]A \cap B = 2[/tex]
15 would not go to the family cottage
[tex]C = 15[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]15 = c + 3 + 3 + 5[/tex]
[tex]c = 4[/tex]
12 would not go to a beach
[tex]B = 12[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]12 = b + 3 + 2 + 5[/tex]
[tex]b = 2[/tex]
11 would not go to a park
[tex]A = 11[/tex]
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]11 = a + 2 + 3 + 5[/tex]
[tex]a = 1[/tex]
Now, we can find the total number of family members.
[tex]T = a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]T = 1 + 2 + 4 + 1 + 2 + 3 + 3 + 5[/tex]
[tex]T = 21[/tex]
The total number of family members is 21.
