Forty-two cities were researched to determine whether they had a protessional sports team, a symphony, or a childrer's museum, of these cties, 21 had a professional sports team, 24 had a symphory, 16 had a children's museum, 15 had a professional sports leam and a symphony, 8 had a professional sports team and a childrer's muceum, 11 had a symphony and a childrer's muceum, and 6 had at tries adivities. Complete parts a) through e) below. a) How many of the cities surveyed had only a professional sports team? 4 (Simpility your ancwer) b) How many of the cities surveyed had a protessional sports team and a symphony, but not a children's museum? 9 (Simplity your answer) c) How many of the cities surveyed had a professional sports team or a symphony? 30 (Simplify your answer) d) How many of the cities surveyed had a professional sports team or a symphony, but not a childrer's museum? \( 17^{7} \) (Simplify your answer.) e) How many of the cries surveyed had exactly two of the activities? \( 16^{\circ} \) (Simplify your answer.)

We are given three sets of activities among 42 cities:  • PS (Professional Sports Team): 21 cities  • SY (Symphony): 24 cities  • CM (Children’s Museum): 16 cities And the overlaps:  • PS ∩ SY = 15  • PS ∩ CM = 8  • SY ∩ CM = 11  • PS ∩ SY ∩ CM = 6 We will use these numbers to answer each part. ────────────────────────────── a) Only a professional sports team "Only PS" means the city has a professional sports team but not a symphony and not a children’s museum. The count is:  Only PS = |PS| – (cities in PS with SY and/or CM)  Since PS ∩ SY and PS ∩ CM both include the triple intersection, we adjust by adding it back once:  Only PS = 21 – (15 + 8) + 6      = 21 – 23 + 6      = 4 Answer: 4 ────────────────────────────── b) Professional sports team and symphony but not a children’s museum This count is:  PS ∩ SY only = |PS ∩ SY| – |PS ∩ SY ∩ CM|      = 15 – 6      = 9 Answer: 9 ────────────────────────────── c) Professional sports team or symphony We want the union of PS and SY:  |PS ∪ SY| = |PS| + |SY| – |PS ∩ SY|       = 21 + 24 – 15       = 30 Answer: 30 ────────────────────────────── d) Professional sports team or symphony, but not a children’s museum We start with the total number of cities with PS or SY (which is 30 from part c) and then exclude those that also have a children’s museum. We find the number of cities in (PS ∪ SY) that also have CM. Note that:  (PS ∪ SY) ∩ CM = (PS ∩ CM) ∪ (SY ∩ CM)  Using the inclusion–exclusion principle:      = 8 + 11 – 6      = 13 Thus, the cities with (PS or SY) but not CM:      = (PS ∪ SY) – ((PS ∪ SY) ∩ CM)      = 30 – 13      = 17 Answer: 17 ────────────────────────────── e) Exactly two of the activities We add the counts for each two-activity combination, excluding those that have all three:  • PS and SY only = 15 – 6 = 9  • PS and CM only = 8 – 6 = 2  • SY and CM only = 11 – 6 = 5 Total exactly two = 9 + 2 + 5 = 16 Answer: 16 ────────────────────────────── Final Answers: a) 4 b) 9 c) 30 d) 17 e) 16