Question

a group of 2431 students were surveyed about the courses they were taking at their college with the following results:
1171 students said they were taking science.
1277 students said they were taking psychology.
1162 students said they were taking dance.
528 students said they were taking psychology and science.
554 students said they were taking dance and science.
635 students said they were taking dance and psychology.
242 students said they were taking all three courses.
fill in the following venn diagram with the cardinality of each region.

Explanation:

Step1: Calculate region II (Dance and Psychology only)

The number of students taking Dance and Psychology is \(635\). The number of students taking all three courses is \(242\). So, the number of students taking only Dance and Psychology (region II) is \(635 - 242=393\).

Step2: Calculate region IV (Dance and Science only)

The number of students taking Dance and Science is \(554\). The number of students taking all three courses is \(242\). So, the number of students taking only Dance and Science (region IV) is \(554 - 242 = 312\).

Step3: Calculate region VI (Psychology and Science only)

The number of students taking Psychology and Science is \(528\). The number of students taking all three courses is \(242\). So, the number of students taking only Psychology and Science (region VI) is \(528-242 = 286\).

Step4: Calculate region I (Dance only)

The number of students taking Dance is \(1162\). Subtract the students in regions II, IV, and V (all three courses). So, region I: \(1162-(393 + 312+242)=1162 - 947=215\).

Step5: Calculate region III (Psychology only)

The number of students taking Psychology is \(1277\). Subtract the students in regions II, VI, and V. So, region III: \(1277-(393+286 + 242)=1277 - 921=356\).

Step6: Calculate region VII (Science only)

The number of students taking Science is \(1171\). Subtract the students in regions IV, VI, and V. So, region VII: \(1171-(312+286+242)=1171 - 840 = 331\).

Step7: Calculate region VIII (Neither)

First, find the number of students in at least one course. Using the principle of inclusion - exclusion: \(n(D\cup P\cup S)=n(D)+n(P)+n(S)-n(D\cap P)-n(D\cap S)-n(P\cap S)+n(D\cap P\cap S)\)
\(n(D\cup P\cup S)=1162 + 1277+1171-635 - 554-528+242\)
\(n(D\cup P\cup S)=1162+1277 + 1171-(635 + 554+528)+242\)
\(n(D\cup P\cup S)=3610 - 1717+242=2135\)
The total number of students is \(2431\). So, region VIII: \(2431-2135 = 296\).

Step8: Region V (All three courses)

Given as \(242\).

Answer:

I. \(215\)
II. \(393\)
III. \(356\)
IV. \(312\)
V. \(242\)
VI. \(286\)
VII. \(331\)
VIII. \(296\)