Question

thirty-three cities were researched to determine whether they had a professional sports team, a symphony, or a children’s museum. of these cities, 16 had a professional sports team, 17 had a symphony, 14 had a children’s museum, 10 had a professional sports team and a symphony, 5 had a professional sports team and a children’s museum, 7 had a symphony and a children’s museum, and 3 had all three activities. complete parts a) through e) below

a) how many of the cities surveyed had only a professional sports team?
4 (simplify your answer.)

b) how many of the cities surveyed had a professional sports team and a symphony, but not a children’s museum?
7 (simplify your answer.)

c) how many of the cities surveyed had a professional sports team or a symphony?
23 (simplify your answer.)

d) how many of the cities surveyed had a professional sports team or a symphony, but not a children’s museum?
14 (simplify your answer.)

e) how many of the cities surveyed had exactly two of the activities?
\boxed{} (simplify your answer.)

Explanation:

Step1: Analyze two - activity overlaps

For professional sports team (S) and symphony (Sy): Only S and Sy (not children's museum - M) is \(10 - 3=7\).
For professional sports team (S) and children's museum (M): Only S and M (not symphony - Sy) is \(5 - 3 = 2\).
For symphony (Sy) and children's museum (M): Only Sy and M (not professional sports team - S) is \(7 - 3=4\).

Step2: Sum the two - activity only counts

To find the number of cities with exactly two activities, we sum the three "only two" groups: \(7 + 2+4\).
\(7+2 = 9\), \(9 + 4=13\).

Answer:

13