The number of people who liked raspberries and blueberries but not strawberries is 18.
Let's break down the information given using a Venn diagram to find the number of people who liked raspberries and blueberries but not strawberries.
Let S represent the set of people who liked strawberries, R for raspberries, and B for blueberries. We're given:
∣S∣=96 (Number of people who liked strawberries)
∣R∣=98 (Number of people who liked raspberries)
∣B∣=120 (Number of people who liked blueberries)
∣S∩R∩B∣=38 (Number of people who liked all three)
We also know:
∣S∩R∣=18 (People who liked both strawberries and raspberries)
∣R∩B∣=? (We're trying to find the number of people who liked raspberries and blueberries but not strawberries)
Using the principle of inclusion-exclusion:
∣S∪R∪B∣=∣S∣+∣R∣+∣B∣−∣S∩R∣−∣R∩B∣−∣S∩B∣+∣S∩R∩B∣
Substitute the given values:
96+98+120−18−∣R∩B∣−∣S∩B∣+38=∣S∪R∪B∣
Simplify:
314−∣R∩B∣−∣S∩B∣=∣S∪R∪B∣
We know the total number of people is the sum of those who like at least one of the fruits:
∣S∪R∪B∣=∣S∣+∣R∣+∣B∣−∣S∩R∣−∣R∩B∣−∣S∩B∣+∣S∩R∩B∣
∣S∪R∪B∣=96+98+120−18−∣R∩B∣−∣S∩B∣+38
∣S∪R∪B∣=296−∣R∩B∣−∣S∩B∣
So,
∣S∪R∪B∣=314−∣R∩B∣−∣S∩B∣
Equating the two expressions for
∣S∪R∪B∣:
314−∣R∩B∣−∣S∩B∣=296−∣R∩B∣−∣S∩B∣
Solve for ∣R∩B∣:
∣R∩B∣=314−296=18
Therefore, the number of people who liked raspberries and blueberries but not strawberries is 18.
Question
In a survey 96 people liked strawberries, 98 liked raspberries, and 120 liked blueberries; 18 people liked only strawberries, 20 liked only raspberries, and 24 liked only blueberries; 38 people liked all three. How many people liked raspberries and blueberries but not strawberries?