Our Indiscrete Mathematics course has21 students from the the College of Arts, 15 of whom are female;26 students from the the College of Engineering and Informatics, 7 of whom are female;31 students from the the College of Science, 12 of whom are female.How many ways can we choose a single class rep?Answer: How many ways can we choose three reps, one from each of the three Colleges?Answer: How many ways can we choose three reps, one from each of the three Colleges, so that exactly one is female?Answer:
Question
Our Indiscrete Mathematics course has21 students from the the College of Arts, 15 of whom are female;26 students from the the College of Engineering and Informatics, 7 of whom are female;31 students from the the College of Science, 12 of whom are female.How many ways can we choose a single class rep?Answer: How many ways can we choose three reps, one from each of the three Colleges?Answer: How many ways can we choose three reps, one from each of the three Colleges, so that exactly one is female?Answer:
Solution
Sure, here are the answers to your questions:
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The total number of students is 21 (Arts) + 26 (Engineering and Informatics) + 31 (Science) = 78 students. So, there are 78 ways to choose a single class representative.
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To choose one representative from each college, we multiply the number of students in each college together. So, there are 21 (Arts) * 26 (Engineering and Informatics) * 31 (Science) = 16826 ways to choose one representative from each college.
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To choose exactly one female representative from the three colleges, we have to consider three scenarios: the female representative comes from Arts, Engineering and Informatics, or Science.
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If the female representative is from Arts, there are 15 choices. For the other two colleges, we choose from the male students, which are 26-7=19 from Engineering and Informatics and 31-12=19 from Science. So there are 15 * 19 * 19 = 5415 ways.
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If the female representative is from Engineering and Informatics, there are 7 choices. For the other two colleges, we choose from the male students, which are 21-15=6 from Arts and 31-12=19 from Science. So there are 7 * 6 * 19 = 798 ways.
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If the female representative is from Science, there are 12 choices. For the other two colleges, we choose from the male students, which are 21-15=6 from Arts and 26-7=19 from Engineering and Informatics. So there are 12 * 6 * 19 = 1368 ways.
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Adding these up, there are 5415 + 798 + 1368 = 7581 ways to choose three representatives so that exactly one is female.
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