Finding the Number of Students Who Like Both Chocolate and Vanilla
Understanding Student Preferences Through Process of Elimination To better understand our students’ tastes and preferences, we surveyed the class to find out how many liked chocolate, vanilla or neither flavor of ice cream. Using the principle of inclusion-exclusion, we can determine the exact number who enjoy both chocolate and vanilla.
Sizing Up the Initial Data
We first gathered some key facts about our class of 35 students: 15 said they liked chocolate ice cream, 13 preferred vanilla, and 10 didn’t care for either flavor. If we added up just the chocolate and vanilla fans, the total would be 28 - larger than the actual class size. This told us some students must overlap in their tastes.
Accounting for Students With Single Preferences
Let’s break down the data step-by-step. We’ll call A the number who like chocolate and B the number who prefer vanilla. We know A is 15 and B is 13. The 10 students indifferent to both flavors can be labeled N. So the total students accounting for single preferences is A + B + N, which equals 15 + 13 + 10 = 38.
Subtracting Students With No Preferences
But our class only has 35 students. So we need to subtract those who don’t care for either flavor (N), which is 10. That leaves A + B + N - N = A + B, which is 15 + 13 = 28.
Calculating the Overlap Between Groups
Now we can use the principle of inclusion-exclusion. The overlap between groups A and B - those who like both chocolate and vanilla - is the difference between the total accounting for single preferences (28) and the actual class size (35). Therefore, A ∩ B = 28 - 35 = 3.
Confirming the Number Who Enjoy Both Flavors
Through a step-by-step logical process of accounting for students based on their stated preferences, and then utilizing the principle of inclusion-exclusion, we can confirm that the number of students in our class who enjoy both chocolate and vanilla ice cream is 3. Understanding our students’ diverse tastes helps us gain valuable insights.
