Question

in a survey of 253 professional athletes, it was found that 104 of them owned a convertible, 113 of them owned a giant screen tv, and 118 owned a sporting goods store. 26 owned a convertible and a store, 51 owned a tv and a store, and 52 owned a convertible and a tv. 12 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:

Let \(C\) be the set of athletes who own a convertible, \(T\) be the set of athletes who own a giant - screen TV, and \(S\) be the set of athletes who own a sporting goods store. We know \(n(C)=104\), \(n(T)=113\), \(n(S)=118\), \(n(C\cap S)=26\), \(n(T\cap S)=51\), \(n(C\cap T)=52\), and \(n(C\cap T\cap S)=12\).

Step1: Use the principle of inclusion - exclusion for \(n(C\cup T\cup S)\)

The formula is \(n(C\cup T\cup S)=n(C)+n(T)+n(S)-n(C\cap T)-n(C\cap S)-n(T\cap S)+n(C\cap T\cap S)\).
\[

$$\begin{align*} n(C\cup T\cup S)&=104 + 113+118-52 - 26-51+12\\ &=217+118-(52 + 26+51)+12\\ &=335 - 129+12\\ &=218 \end{align*}$$

\]

Step2: Calculate the number of athletes who own none of the three items

The total number of athletes surveyed is \(N = 253\). The number of athletes who own none of the three items is \(253 - n(C\cup T\cup S)\).
\[253-218 = 35\]

Step3: Calculate the number of athletes who own a convertible and a TV but not a store

We know \(n(C\cap T)=52\) and \(n(C\cap T\cap S)=12\). So the number is \(n(C\cap T)-n(C\cap T\cap S)=52 - 12=40\)

Step4: Calculate the number of athletes who own a convertible or a TV

Using the inclusion - exclusion principle for \(n(C\cup T)\), \(n(C\cup T)=n(C)+n(T)-n(C\cap T)=104 + 113-52=165\)

Step5: Calculate the number of athletes who own exactly one type of item

Number of athletes who own only a convertible: \(n(C)-n(C\cap T)-n(C\cap S)+n(C\cap T\cap S)=104-52 - 26+12 = 38\)
Number of athletes who own only a TV: \(n(T)-n(C\cap T)-n(T\cap S)+n(C\cap T\cap S)=113-52 - 51+12 = 22\)
Number of athletes who own only a store: \(n(S)-n(C\cap S)-n(T\cap S)+n(C\cap T\cap S)=118-26 - 51+12 = 53\)
The number of athletes who own exactly one type of item is \(38 + 22+53=113\)

Step6: Calculate the number of athletes who own at least one type of item

This is \(n(C\cup T\cup S)=218\)

Step7: Calculate the number of athletes who own a TV or a store but not a convertible

First, \(n((T\cup S)\cap\overline{C})=n(T\cup S)-n((T\cup S)\cap C)\)
We know \(n(T\cup S)=n(T)+n(S)-n(T\cap S)=113 + 118-51 = 180\)
\(n((T\cup S)\cap C)=n((T\cap C)\cup(S\cap C))=n(T\cap C)+n(S\cap C)-n(T\cap S\cap C)=52+26 - 12=66\)
So \(n((T\cup S)\cap\overline{C})=180-66 = 114\)

Answer:

1. 35
2. 40
3. 165
4. 113
5. 218
6. 114