Questions: An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 815 drank alcohol regularly, 626 smoked cigarettes, 183 used illegal drugs, 394 drank alcohol regularly and smoked cigarettes, 100 drank alcohol regularly and used illegal drugs, 111 smoked cigarettes and used illegal drugs, 81 engaged in all three behaviors, and 150 engaged in none of these behaviors. a) How many students were surveyed? 1250 Of those surveyed, b) How many drank alcohol or smoked cigarettes? 1047 c) How many drank alcohol only? 402 ? d) How many drank alcohol and smoked cigarettes, but did not use illegal drugs? 313 e) How many smoked cigarettes or used illegal drugs, but did not drink alcohol? 285? f) How many engaged in exactly two of these behaviors?

An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 815 drank alcohol regularly, 626 smoked cigarettes, 183 used illegal drugs, 394 drank alcohol regularly and smoked cigarettes, 100 drank alcohol regularly and used illegal drugs, 111 smoked cigarettes and used illegal drugs, 81 engaged in all three behaviors, and 150 engaged in none of these behaviors. a) How many students were surveyed? 1250 Of those surveyed, b) How many drank alcohol or smoked cigarettes? 1047 c) How many drank alcohol only? 402 ? d) How many drank alcohol and smoked cigarettes, but did not use illegal drugs? 313 e) How many smoked cigarettes or used illegal drugs, but did not drink alcohol? 285? f) How many engaged in exactly two of these behaviors?
Transcript text: An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 815 drank alcohol regularly, 626 smoked cigarettes, 183 used illegal drugs, 394 drank alcohol regularly and smoked cigarettes, 100 drank alcohol regularly and used illegal drugs, 111 smoked cigarettes and used illegal drugs, 81 engaged in all three behaviors, and 150 engaged in none of these behaviors. a) How many students were surveyed? 1250 Of those surveyed, b) How many drank alcohol or smoked cigarettes? $1047^{7}$ c) How many drank alcohol only? 402 ? d) How many drank alcohol and smoked cigarettes, but did not use illegal drugs? 313 e) How many smoked cigarettes or used illegal drugs, but did not drink alcohol? 285? f) How many engaged in exactly two of these behaviors? $\square$
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Solution

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Solution Steps

Step 1: Total Number Surveyed

To find the total number of individuals surveyed, we use the principle of inclusion-exclusion. The formula is: \[ Total = A + B + C - AB - AC - BC + ABC + N = 815 + 626 + 183 - 394 - 100 - 111 + 81 + 150 = 1250 \]

Step 2: Specific Combinations

For individuals engaging in exactly one activity: \[ A_{only} = A - AB - AC + ABC = 815 - 394 - 100 + 81 = 402 \] \[ B_{only} = B - AB - BC + ABC = 626 - 394 - 111 + 81 = 202 \] \[ C_{only} = C - AC - BC + ABC = 183 - 100 - 111 + 81 = 53 \] For individuals engaging in exactly two activities: \[ AB_{only} = AB - ABC = 394 - 81 = 313 \] \[ AC_{only} = AC - ABC = 100 - 81 = 19 \] \[ BC_{only} = BC - ABC = 111 - 81 = 30 \] For individuals engaging in all three activities: \[ ABC = 81 \] For individuals engaging in none of the activities: \[ N = 150 \]

Step 3: Engaged in At Least One Activity

To find the number of individuals engaged in at least one activity, subtract the number of individuals engaging in none of the activities from the total surveyed. \[ AtLeastOne = Total - N = 1250 - 150 = 1100 \]

Final Answer:

The total number of individuals surveyed is 1250. Number of individuals engaging in exactly one activity: A only = 402, B only = 202, C only = 53. Number of individuals engaging in exactly two activities: AB only = 313, AC only = 19, BC only = 30. Number of individuals engaging in all three activities: 81. Number of individuals engaging in none of the activities: 150. Number of individuals engaged in at least one activity: 1100.

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