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.
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Welcome back bishalsinjalimagar496 — missed you this week. The image contains multiple questions. I will solve question b) based on the provided image. b) In a survey of a group, 60 like tea, 45 like coffee, 30 like milk, 25 like coffee and tea, 20 like milk and tea, 15 like coffee and milk and 10 like all three drinks. Based on the information, answer the following questions: i) Show the information in a Venn-diagram. ii) Find how many people were surveyed. Let T be the set of people who like tea, C be the set of people who like coffee, and M be the set of people who like milk. Given: n(T) = 60 n(C) = 45 n(M) = 30 n(C T) = 25 n(M T) = 20 n(C M) = 15 n(C T M) = 10 Step 1: Calculate the number of people who like exactly two drinks. • Only Coffee and Tea: n(C T) - n(C T M) = 25 - 10 = 15 • Only Milk and Tea: n(M T) - n(C T M) = 20 - 10 = 10 • Only Coffee and Milk: n(C M) - n(C T M) = 15 - 10 = 5 Step 2: Calculate the number of people who like only one drink. • Only Tea: n(T) - [n(C T) - n(C T M)] - [n(M T) - n(C T M)] - n(C T M) n(T) - n(C T) - n(M T) + n(C T M) + n(C T M) - n(C T M) n(T) - n(C T) - n(M T) + n(C T M) 60 - 15 - 10 - 10 = 60 - (15+10+10) = 60 - 35 = 25 • Only Coffee: n(C) - [n(C T) - n(C T M)] - [n(C M) - n(C T M)] - n(C T M) n(C) - n(C T) - n(C M) + n(C T M) 45 - 15 - 5 - 10 = 45 - (15+5+10) = 45 - 30 = 15 • Only Milk: n(M) - [n(M T) - n(C T M)] - [n(C M) - n(C T M)] - n(C T M) n(M) - n(M T) - n(C M) + n(C T M) 30 - 10 - 5 - 10 = 30 - (10+5+10) = 30 - 25 = 5 Step 3: i) Show the information in a Venn-diagram. The Venn diagram would show three overlapping circles for Tea (T), Coffee (C), and Milk (M). • The intersection of all three (T C M) is 10. • The intersection of only T and C is 15. • The intersection of only T and M is 10. • The intersection of only C and M is 5. • The region for only T is 25. • The region for only C is 15. • The region for only M is 5. c Venn Diagram: \\ Total surveyed \\ T (Tea) C (Coffee) M (Milk) \\ Only T: 25 \\ Only C: 15 \\ Only M: 5 \\ Only T and C: 15 \\ Only T and M: 10 \\ Only C and M: 5 \\ T and C and M: 10 Step 4: ii) Find how many people were surveyed. The total number of people surveyed is the union of all three sets, n(T C M). Using the Principle of Inclusion-Exclusion: n(T C M) = n(T) + n(C) + n(M) - n(T C) - n(T M) - n(C M) + n(T C M) n(T C M) = 60 + 45 + 30 - 25 - 20 - 15 + 10 n(T C M) = 135 - 60 + 10 n(T C M) = 75 + 10 n(T C M) = 85 Alternatively, sum all the distinct regions from Step 1 and Step 2: n(T C M) = (Only T) + (Only C) + (Only M) + (Only T and C) + (Only T and M) + (Only C and M) + (T and C and M) n(T C M) = 25 + 15 + 5 + 15 + 10 + 5 + 10 n(T C M) = 85 The number of people surveyed is 85. Drop the next question.