Question

In a survey of 203 professional athletes, it was fcund that 96 of them owned a convertible, 86 of them owned a giant screen TV, and 98 owited a sporting goods store. 25 owned a convertible and a store, 35 owned a TV and a store, and 51 owned a convertable and a TV. 15 owned all three items. a. How many athletes did not own any of the three items?

Answer 1

To find the number of athletes who did not own any of the three items, we can use the principle of inclusion-exclusion. By adding up the individual counts and subtracting the number of athletes who owned all three items, we can calculate the number of athletes who owned at least one item. Subtracting this number from the total number of athletes surveyed gives us the number of athletes who did not own any of the three items. In this case, there are 29 athletes who did not own any of the three items.

Step-by-step explanation:

To find the number of athletes who did not own any of the three items, we need to subtract the number of athletes who owned at least one item from the total number of athletes surveyed.

We can use the principle of inclusion-exclusion to calculate the number of athletes who owned at least one item.

1. There are 96 athletes who owned a convertible.
2. There are 86 athletes who owned a giant screen TV.
3. There are 98 athletes who owned a sporting goods store.
4. There are 25 athletes who owned a convertible and a store.
5. There are 35 athletes who owned a TV and a store.
6. There are 51 athletes who owned a convertible and a TV.
7. There are 15 athletes who owned all three items.

To calculate the number of athletes who owned at least one item, we add up the individual counts and subtract the number of athletes who owned all three items:

Number of athletes who owned at least one item = 96 + 86 + 98 - 25 - 35 - 51 + 15 = 174

Therefore, the number of athletes who did not own any of the three items would be:

Number of athletes who did not own any of the three items = Total number of athletes surveyed - Number of athletes who owned at least one item.

Number of athletes who did not own any of the three items = 203 - 174 = 29.

So, there are 29 athletes who did not own any of the three items.