Answer:
3 of the cities surveyed had only a professional sports team.
Step-by-step explanation:
We solve this question treating these as Venn sets.
I am going to say that:
Set A: professional sports team.
Set B: Symphony.
Set C: Children's museum.
We start building from the intersection of these three.
6 had all three activities.
This means that ![A \cap B \cap C = 6](https://tex.z-dn.net/?f=A%20%5Ccap%20B%20%5Ccap%20C%20%3D%206)
10 had a symphony and a children's museum,
This means that:
![(B \cap C) + (A \cap B \cap C) = 10](https://tex.z-dn.net/?f=%28B%20%5Ccap%20C%29%20%2B%20%28A%20%5Ccap%20B%20%5Ccap%20C%29%20%3D%2010)
![(B \cap C) + 6 = 10](https://tex.z-dn.net/?f=%28B%20%5Ccap%20C%29%20%2B%206%20%3D%2010)
![(B \cap C) = 4](https://tex.z-dn.net/?f=%28B%20%5Ccap%20C%29%20%3D%204)
That is, 4 have only the symphony and children's museum, while 6 have all three.
10 had a professional sports team and a children's museum
![(A \cap C) + (A \cap B \cap C) = 10](https://tex.z-dn.net/?f=%28A%20%5Ccap%20C%29%20%2B%20%28A%20%5Ccap%20B%20%5Ccap%20C%29%20%3D%2010)
![(A \cap C) + 6 = 10](https://tex.z-dn.net/?f=%28A%20%5Ccap%20C%29%20%2B%206%20%3D%2010)
![(A \cap C) = 4](https://tex.z-dn.net/?f=%28A%20%5Ccap%20C%29%20%3D%204)
14 had a professional sports team and a symphony
![(A \cap B) + (A \cap B \cap C) = 14](https://tex.z-dn.net/?f=%28A%20%5Ccap%20B%29%20%2B%20%28A%20%5Ccap%20B%20%5Ccap%20C%29%20%3D%2014)
![(A \cap B) + 6 = 14](https://tex.z-dn.net/?f=%28A%20%5Ccap%20B%29%20%2B%206%20%3D%2014)
![(A \cap B) = 8](https://tex.z-dn.net/?f=%28A%20%5Ccap%20B%29%20%3D%208)
21 had a professional sports team
This means that:
![(A - B - C) + (A \cap B) + (A \cap C) + (A \cap B \cap C) = 21](https://tex.z-dn.net/?f=%28A%20-%20B%20-%20C%29%20%2B%20%28A%20%5Ccap%20B%29%20%2B%20%28A%20%5Ccap%20C%29%20%2B%20%28A%20%5Ccap%20B%20%5Ccap%20C%29%20%3D%2021)
In which (A - B - C) represents the cities which had only a sports team. So
![(A - B - C) + 8 + 4 + 6 = 21](https://tex.z-dn.net/?f=%28A%20-%20B%20-%20C%29%20%2B%208%20%2B%204%20%2B%206%20%3D%2021)
![(A - B - C) = 3](https://tex.z-dn.net/?f=%28A%20-%20B%20-%20C%29%20%3D%203)
3 of the cities surveyed had only a professional sports team.