Question

in a survey of 274 professional athletes, it was found that 136 of them owned a convertible, 121 of them owned a giant screen tv, and 106 owned a sporting goods store. 20 owned a convertible and a store, 45 owned a tv and a store, and 70 owned a convertible and a tv. 13 owned all three items.

1. how many athletes did not own any of the three items?
2. how many owned a convertible and a tv, but not a store?
3. how many athletes owned a convertible or a tv?
4. how many athletes owned exactly one type of item in the survey?
5. how many athletes owned at least one type of item in the survey?
6. how many owned a tv or a store, but not a convertible?

Explanation:

Step1: Define sets and values

Let \(A\) be the set of athletes who own a convertible (\(|A| = 136\)), \(B\) be the set of athletes who own a TV (\(|B|=121\)), and \(C\) be the set of athletes who own a store (\(|C| = 106\)). Also, \(|A\cap B| = 70\), \(|A\cap C|=20\), \(|B\cap C| = 45\), and \(|A\cap B\cap C|=13\). The total number of athletes \(N = 274\).

Step2: Use the principle of inclusion - exclusion

The formula for \(|A\cup B\cup C|\) is \(|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|\).
Substitute the values: \(136 + 121+106-70 - 20-45 + 13\).
First, add the individual - set sizes: \(136+121 + 106=363\).
Then, add the sizes of the two - set intersections: \(70 + 20+45 = 135\).
So, \(|A\cup B\cup C|=363-135 + 13=241\).

Step3: Answer question 1

The number of athletes who did not own any of the three items is \(N-|A\cup B\cup C|\). So, \(274 - 241=33\).

Step4: Answer question 2

The number of athletes who owned a convertible and a TV but not a store is \(|A\cap B|-|A\cap B\cap C|\). So, \(70 - 13=57\).

Step5: Answer question 3

The number of athletes who owned a convertible or a TV is \(|A\cup B|\). Using the formula \(|A|+|B|-|A\cap B|\), we have \(136+121 - 70=187\).

Step6: Answer question 4

The number of athletes who owned only a convertible is \(|A|-(|A\cap B|+|A\cap C|)+|A\cap B\cap C|=136-(70 + 20)+13=59\).
The number of athletes who owned only a TV is \(|B|-(|A\cap B|+|B\cap C|)+|A\cap B\cap C|=121-(70 + 45)+13=19\).
The number of athletes who owned only a store is \(|C|-(|A\cap C|+|B\cap C|)+|A\cap B\cap C|=106-(20 + 45)+13=54\).
The number of athletes who owned exactly one type of item is \(59+19 + 54=132\).

Step7: Answer question 5

The number of athletes who owned at least one type of item is \(|A\cup B\cup C| = 241\).

Step8: Answer question 6

The number of athletes who owned a TV or a store but not a convertible is \((|B|+|C|-|B\cap C|)-(|A\cap B|+|A\cap C|)+|A\cap B\cap C|\).
First, \(|B|+|C|-|B\cap C|=121 + 106-45=182\).
\(|A\cap B|+|A\cap C|=70 + 20=90\).
So, \(182-90+13=105\).

Answer:

1. 33
2. 57
3. 187
4. 132
5. 241
6. 105