Question

a group of 2392 students were surveyed about the courses they were taking at their college with the following results: 1153 students said they were taking science. 1259 students said they were taking history. 1259 students said they were taking dance. 602 students said they were taking dance and history. 577 students said they were taking dance and science. 551 students said they were taking history and science. 227 students said they were taking all three courses. fill in the following venn diagram with the cardinality of each region then answer the questions below. how many students took dance, history, or science? how many students took dance or didnt take history? how many students took history and science, but not dance? how many students took dance or science, but not history? how many students took none of the courses?

Explanation:

Step1: Calculate region I

Students taking Dance and History is 602 and all - three - course students is 227. So students taking only Dance and History (region II) is \(602 - 227=375\). Students taking Dance and Science is 577 and all - three - course students is 227. So students taking only Dance and Science (region IV) is \(577 - 227 = 350\). Students taking Dance is 1259. So region I (only Dance) is \(1259-(375 + 227+350)=1259 - 952 = 307\).

Step2: Calculate region III

Students taking History and Science is 551 and all - three - course students is 227. So students taking only History and Science (region VI) is \(551 - 227=324\). Students taking History is 1259. So region III (only History) is \(1259-(375 + 227+324)=1259 - 926 = 333\).

Step3: Calculate region VII

Students taking Science is 1153. So region VII (only Science) is \(1153-(350 + 227+324)=1153 - 901 = 252\).

Step4: Calculate number of students taking Dance, History or Science

Using the principle of inclusion - exclusion \(n(D\cup H\cup S)=n(D)+n(H)+n(S)-n(D\cap H)-n(D\cap S)-n(H\cap S)+n(D\cap H\cap S)\)
\(n(D\cup H\cup S)=1259 + 1259+1153-602 - 577-551+227=2168\).

Step5: Calculate number of students taking Dance or didn't take History

Number of students taking Dance is 1259. Number of students not taking History is \(2392 - 1259 = 1133\). Number of students taking Dance and History is 602. So \(n(D\cup\overline{H})=1259+1133 - 602=1790\).

Step6: Calculate number of students taking History and Science but not Dance

Region VI is 324.

Step7: Calculate number of students taking Dance or Science but not History

Region I + Region IV+ Region VII \(=307+350 + 252=909\).

Step8: Calculate number of students taking none of the courses

Total students is 2392 and students taking Dance, History or Science is 2168. So number of students taking none of the courses is \(2392-2168 = 224\).

Answer:

Region I: 307
Region II: 375
Region III: 333
Region IV: 350
Region V: 227
Region VI: 324
Region VII: 252
Region VIII: 224
Number of students taking Dance, History, or Science: 2168
Number of students taking Dance or didn't take History: 1790
Number of students taking History and Science, but not Dance: 324
Number of students taking Dance or Science, but not History: 909
Number of students taking none of the courses: 224