in a survey of 274 professional athletes, it was found that 136 of them owned a convertible, 121 of them…

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.\n1. how many athletes did not own any of the three items?\n2. how many owned a convertible and a tv, but not a store?\n3. how many athletes owned a convertible or a tv?\n4. how many athletes owned exactly one type of item in the survey?\n5. how many athletes owned at least one type of item in the survey?\n6. how many owned a tv or a store, but not a convertible?

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.\n1. how many athletes did not own any of the three items?\n2. how many owned a convertible and a tv, but not a store?\n3. how many athletes owned a convertible or a tv?\n4. how many athletes owned exactly one type of item in the survey?\n5. how many athletes owned at least one type of item in the survey?\n6. how many owned a tv or a store, but not a convertible?

Answer

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