An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 851 drank alcohol regularly, 648 smoked cigarettes, 200 used illegal drugs, 416 drank alcohol regularly and smoked cigarettes, 108 drank alcohol regularly and used illegal drugs, 112 smoked cigarettes and used illegal drugs, engaged in all three behaviors, and 248 engaged in none of these behaviors.
a) How many students were surveyed?
Answer
Final Answer: The total number of students surveyed is \(\boxed{1311}\).
Step 1 ：An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 851 drank alcohol regularly, 648 smoked cigarettes, 200 used illegal drugs, 416 drank alcohol regularly and smoked cigarettes, 108 drank alcohol regularly and used illegal drugs, 112 smoked cigarettes and used illegal drugs, engaged in all three behaviors, and 248 engaged in none of these behaviors.
Step 2 ：To find the total number of students surveyed, we need to add up all the individual groups of students. However, we need to be careful not to double count students who are in more than one group. For example, the 416 students who both drink alcohol regularly and smoke cigarettes are already included in the 851 who drink alcohol and the 648 who smoke cigarettes. So, we need to subtract these overlapping groups to avoid double counting. The same goes for the other overlapping groups. However, the question does not provide the number of students who engaged in all three behaviors. We need this information to accurately calculate the total number of students surveyed.
Step 3 ：Let's denote the number of students who drink alcohol as \(a\), smoke cigarettes as \(c\), use illegal drugs as \(d\), drink alcohol and smoke cigarettes as \(ac\), drink alcohol and use illegal drugs as \(ad\), smoke cigarettes and use illegal drugs as \(cd\), and those who do not engage in any of these behaviors as \(n\).
Step 4 ：Then, the total number of students surveyed can be calculated as \(a + c + d - ac - ad - cd + n\).
Step 5 ：Substituting the given values, we get \(851 + 648 + 200 - 416 - 108 - 112 + 248 = 1311\).
Step 6 ：Final Answer: The total number of students surveyed is \(\boxed{1311}\).
