Question

The number of students who opted any three of the four subjects is

• A. \(62\)
• B. \(63\)
• C. \(64\)
• D. \(66\)
• A. \(320\)
• B. \(300\)
• C. \(210\)
• D. \(183\)
• A. \(72\)
• B. \(65\)
• C. \(63\)
• D. \(61\)
• A. Mathematics
• B. Geography
• C. Physics
• D. History

Answer 1

The Correct Option is A

Concept: “Any three subjects” means students who opted exactly three subjects. So we add all the regions where exactly three circles overlap. Do not include the center (all four subjects).

Step 1: Identify the regions of exactly three subjects.
From the Venn diagram: \[ M \cap H \cap G = 13 \] \[ M \cap H \cap P = 18 \] \[ H \cap G \cap P = 18 \] \[ M \cap G \cap P = 13 \]

Step 2: Add them.
\[ 13+18+18+13 \] \[ =62 \] Thus, the number of students who opted exactly three subjects is: \[ \boxed{62} \]

Answer 2

Concept: “History or Mathematics or Physics” means all students belonging to at least one of these three sets. So we include all regions covered by these three circles. Only the Geography-only region is excluded.

Step 1: List all relevant regions.
History only: \[ 16 \] Mathematics only: \[ 9 \] Physics only: \[ 19 \] Math \(\cap\) History: \[ 14 \] History \(\cap\) Geography: \[ 12 \] Math \(\cap\) Physics: \[ 15 \] Physics \(\cap\) Geography: \[ 16 \] Math \(\cap\) History \(\cap\) Geography: \[ 13 \] Math \(\cap\) History \(\cap\) Physics: \[ 18 \] History \(\cap\) Geography \(\cap\) Physics: \[ 18 \] Math \(\cap\) Physics \(\cap\) Geography: \[ 13 \] All four: \[ 20 \]

Step 2: Add them.
\[ 16+9+19+14+12+15+16+13+18+18+13+20 \] \[ =183 \] Thus, the total number of students is: \[ \boxed{183} \]

Answer 3

Concept: “Both History and Geography” means all regions common to History and Geography. This includes: - only History \(\cap\) Geography - triple intersections involving both - all four common intersection

Step 1: Identify the common regions.
History \(\cap\) Geography only: \[ 12 \] Math \(\cap\) History \(\cap\) Geography: \[ 13 \] History \(\cap\) Geography \(\cap\) Physics: \[ 18 \] All four subjects: \[ 20 \]

Step 2: Add them.
\[ 12+13+18+20 \] \[ =63 \] Thus, the number of students who opted both History and Geography is: \[ \boxed{63} \]

Answer 4

Concept: To find the subject opted by the largest number of students, add all regions belonging to each subject separately.

Step 1: Find total students in Mathematics.
Mathematics includes: \[ 9+14+15+18+13+13+20 \] \[ =102 \]

Step 2: Find total students in Geography.
Geography includes: \[ 9+12+16+13+18+13+20 \] \[ =101 \]

Step 3: Find total students in History.
History includes: \[ 16+14+12+18+13+18+20 \] \[ =111 \]

Step 4: Find total students in Physics.
Physics includes: \[ 19+15+16+18+13+18+20 \] \[ =119 \]

Step 5: Compare totals.
\[ Mathematics=102 \] \[ Geography=101 \] \[ History=111 \] \[ Physics=119 \] Largest is: \[ 119 \] Thus, the subject opted by the maximum students is: \[ \boxed{\text{Physics}} \]