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:

In which a are those that would only not go to a park,
are those who would not got to a park or to the beach,
are those who would not go to a park or to the famili cottage. And
are those that would not go to any of these places.
By the same logic, we have:


This diagram has the following values:

The total number of family members is the sum of all these values:

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 
1 would go to all three places. This means that
.
8 would go to neither a park nor the family cottage
This means that:


8 would go to neither a beach nor the family cottage


7 would go to neither a park nor a beach


15 would not go to the family cottage




12 would not go to a beach




11 would not go to a park




Now, we can find the total number of family members.



The total number of family members is 21.