Question

In a survey of 273 professional athletes, it was found that 122 of them owned a convertible, 116 of them owned a giant screen TV, and 121 owned a sporting goods store. 24 owned a convertible and a store, 43 owned a TV and a store, and 56 owned a covertible and a TV. 8 owned all three items.

a) How many athletes owned only a convertible?

1) 24
2) 122
3) 8
4) 264

Answer 1

Using the principle of inclusion-exclusion, we calculate that 50 athletes owned only a convertible, which is not listed in the options suggesting a possible error in the provided choices.

Step-by-step explanation:

To determine how many athletes owned only a convertible, we need to apply the principle of inclusion-exclusion for the three sets provided:

• Total number of athletes owning a convertible: 122
• Total number of athletes owning a sporting goods store: 121
• Total number of athletes owning a giant screen TV: 116
• Athletes owning both a convertible and a store: 24
• Athletes owning both a TV and a store: 43
• Athletes owning both a convertible and a TV: 56
• Athletes owning all three items: 8

To calculate the number of athletes who own only a convertible, we take the total number of convertible owners and subtract the number of athletes who own a convertible and at least one of the other two items:

Only convertible owners = Total convertible owners - (Convertible & Store) - (Convertible & TV) + (All three, because they were subtracted twice)

Only convertible owners = 122 - 24 - 56 + 8

Only convertible owners = 50

Therefore, the number of athletes who owned only a convertible is 50, which is not listed as an option among the choices provided, indicating a possible error in the options listed in the question. The correct answer, which is 50, suggests there may have been a typo or a miscalculation in the options given.