This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back Princess — been a while! Let's pick up where you left off. Let T be the set of people who like tea, C for coffee, and J for juice. We are given: Total people = 200 |T| = 120 |C| = 90 |J| = 70 |T ∩ C| = 50 |T ∩ J| = 40 |C ∩ J| = 30 |T ∩ C ∩ J| = 20 (people who like all three) We need to find the number of people who do not like any of the beverages. First, we'll find the number of people who like at least one beverage using the Principle of Inclusion-Exclusion. Step 1: Calculate the number of people who like at least one beverage. The formula for the union of three sets is: |T ∪ C ∪ J| = |T| + |C| + |J| - |T ∩ C| - |T ∩ J| - |C ∩ J| + |T ∩ C ∩ J| Substitute the given values: |T ∪ C ∪ J| = 120 + 90 + 70 - 50 - 40 - 30 + 20 |T ∪ C ∪ J| = 280 - 120 + 20 |T ∪ C ∪ J| = 160 + 20 |T ∪ C ∪ J| = 180 people So, 180 people like at least one of the beverages. Step 2: Calculate the number of people who do not like any beverage. Number of people who do not like any beverage = Total people - |T ∪ C ∪ J| Number of people who do not like any beverage = 200 - 180 Number of people who do not like any beverage = 20 people Step 3: Venn Diagram Representation To visualize this with a Venn diagram, we fill in the regions starting from the innermost intersection: • People who like all three (T ∩ C ∩ J): 20 • People who like only Tea and Coffee (T ∩ C only): |T ∩ C| - |T ∩ C ∩ J| = 50 - 20 = 30 • People who like only Tea and Juice (T ∩ J only): |T ∩ J| - |T ∩ C ∩ J| = 40 - 20 = 20 • People who like only Coffee and Juice (C ∩ J only): |C ∩ J| - |T ∩ C ∩ J| = 30 - 20 = 10 • People who like only Tea: |T| - (T ∩ C only) - (T ∩ J only) - (T ∩ C ∩ J) = 120 - 30 - 20 - 20 = 50 • People who like only Coffee: |C| - (T ∩ C only) - (C ∩ J only) - (T ∩ C ∩ J) = 90 - 30 - 10 - 20 = 30 • People who like only Juice: |J| - (T ∩ J only) - (C ∩ J only) - (T ∩ C ∩ J) = 70 - 20 - 10 - 20 = 20 Summing these individual regions gives the total number of people who like at least one beverage: 20 + 30 + 20 + 10 + 50 + 30 + 20 = 180 people. The number of people who do not like any beverage is the total number of people minus those who like at least one: 200 - 180 = 20 people. What's next? 📸