Final answer:
The question relates to Venn diagrams and the inclusion-exclusion principle in mathematics. By computing the given percentages of 2000 employees' preferences, we determine that 300 employees prefer coffee, tea, and smoking, which corresponds to option (b).
Step-by-step explanation:
The subject of this question is mathematics, specifically involving Venn diagrams and the inclusion-exclusion principle. Step 1: Analyze the given information. We have a total of 2,000 employees. According to the percentages given, 48% preferred coffee (C), which is 960 employees; 54% liked tea (T), which is 1080 employees; and 64% used to smoke (S), equating to 1280 employees. Step 2: Use the inclusion-exclusion principle. The principle helps to determine the number of employees who prefer exactly one, exactly two, or all three of the items. 28% used C and T, which is 560 employees. Step 3: Calculate Those Who Have All Three.
Let the number of employees who prefer all three be X. Using the inclusion-exclusion principle, we can set up the equation: 960 + 1080 + 1280 - (560 + 640 + 600) + X = 2000 - 120. After solving for X, we find that the number of employees who prefer all three (C, T, and S) is 300. To find the number of people who preferred all three options (coffee, tea, and smoking), we can use the principle of inclusion-exclusion. Let's start by finding the number of people who preferred only coffee (C), only tea (T), and only smoking (S). Number who preferred only C = 48% - 28% - 30% + 6% = 26%. Number of people who preferred only T = 54% - 28% - 32% + 6% = 0%. Number of people who preferred only S = 64% - 32% - 30% + 6% = 8%. Since the percentage of people who preferred only T is 0%, it means that everyone who preferred T also preferred either C, S, or both. Therefore, the number of people who preferred all three (C, T, and S) is equal to the number who preferred T, which is 0% of the total number of employees. So, the answer is (a) 240.