Let's analyze the information provided and break it down step by step to fill in the Venn diagram correctly.

Information Given:

• Total number of students surveyed = 2431
• $|H|$ = 1123 (students taking History)
• $|S|$ = 1374 (students taking Science)
• $|D|$ = 1340 (students taking Dance)
• $|H \cap D|$ = 605 (students taking both History and Dance)
• $|S \cap D|$ = 692 (students taking both Science and Dance)
• $|H \cap S|$ = 627 (students taking both History and Science)
• $|H \cap S \cap D|$ = 317 (students taking all three courses)

Define the regions in the Venn Diagram:

• Region I: Only History
• Region II: History and Science, but not Dance
• Region III: Only Science
• Region IV: History and Dance, but not Science
• Region V: All three courses (History, Science, and Dance)
• Region VI: Science and Dance, but not History
• Region VII: Only Dance
• Region VIII: Outside the three circles (students not taking any of these subjects)

Steps to fill the diagram:

1. Region V (History, Science, and Dance) = $H \cap S \cap D = 317$

   • This represents the intersection of all three subjects.

2. Region II (History and Science, but not Dance):

   • $|H \cap S| = 627$
   • Subtract the students in all three subjects from this: $627 - 317 = 310$

3. Region IV (History and Dance, but not Science):

   • $|H \cap D| = 605$
   • Subtract the students in all three subjects from this: $605 - 317 = 288$

4. Region VI (Science and Dance, but not History):

   • $|S \cap D| = 692$
   • Subtract the students in all three subjects from this: $692 - 317 = 375$

5. Region I (Only History):

   • $|H| = 1123$
   • Subtract the students in all overlapping regions related to History: $1123 - (310 + 288 + 317) = 208$

6. Region III (Only Science):

   • $|S| = 1374$
   • Subtract the students in all overlapping regions related to Science: $1374 - (310 + 375 + 317) = 372$

7. Region VII (Only Dance):

   • $|D| = 1340$
   • Subtract the students in all overlapping regions related to Dance: $1340 - (288 + 375 + 317) = 360$

8. Region VIII (Outside all three subjects):

   • Total students surveyed = 2431
   • Subtract the students taking at least one subject (all regions inside the circles): $2431 - (208 + 310 + 372 + 288 + 317 + 375 + 360) = 201$

Final Values for Each Region:

• Region I (Only History) = 208
• Region II (History and Science, but not Dance) = 310
• Region III (Only Science) = 372
• Region IV (History and Dance, but not Science) = 288
• Region V (All three courses) = 317
• Region VI (Science and Dance, but not History) = 375
• Region VII (Only Dance) = 360
• Region VIII (Outside all subjects) = 201

Would you like to verify these values or have further explanations?

Related Questions:

1. How would the Venn diagram change if an additional subject, like Art, was introduced?
2. What is the formula for calculating the number of students not taking any subject based on the given data?
3. How can you express the cardinality of each set in terms of unions and intersections?
4. How does this problem illustrate the principle of inclusion-exclusion?
5. What would happen to the region sizes if the total number of students increased by 10% but the subject distribution remained the same?

Tip:

Always subtract intersections carefully to avoid double-counting overlapping regions when solving Venn diagram problems.