A survey of 138 college students was taken to determine the musical styles they liked. Of those, 35 students listened to techno, 46 to hip hop, and 45 to rap. Also, 18 students listened to techno and rap, 13 to techno and hip hop, and 26 to hip hop and rap. Finally, 8 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions.
a. How many listened to only techno music?
$\mathrm{n}($ only techno) $=$
Answer
Final Answer: The number of students who only listened to techno music is \(\boxed{12}\).
Step 1 ：A survey of 138 college students was taken to determine the musical styles they liked. Of those, 35 students listened to techno, 46 to hip hop, and 45 to rap. Also, 18 students listened to techno and rap, 13 to techno and hip hop, and 26 to hip hop and rap. Finally, 8 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions.
Step 2 ：To find the number of students who only listened to techno music, we need to subtract the number of students who listened to techno and another genre from the total number of students who listened to techno.
Step 3 ：The total number of students who listened to techno is 35. The number of students who listened to techno and another genre is the sum of the students who listened to techno and rap (18), techno and hip hop (13), and all three genres (8).
Step 4 ：So, the number of students who only listened to techno music is 35 - (18 + 13 + 8).
Step 5 ：The result is negative, which is not possible in this context. I must have made a mistake in my calculations. I realized that I double-counted the students who listened to all three genres. They were already included in the counts of students who listened to techno and rap, and techno and hip hop. So, I should only subtract them once from the total number of students who listened to techno.
Step 6 ：I now know the final answer.
Step 7 ：Final Answer: The number of students who only listened to techno music is \(\boxed{12}\).
