Out of a class of 100 students, 25 play at least cricket and football, 15 play at least cricket and hockey, 12 play at least football and hockey and 10 play all the three sports. The number of students playing cricket, football and hockey are 50, 37, and 22, respectively. The number of students who do NOT play any of the three sports is

The Correct answer is Option 1. 

Key PointsTo find the number of students who do not play any of the three sports (cricket, football, and hockey), we can use the principle of inclusion-exclusion.

Step 1: Define the sets

• C = number of students playing cricket = 50
• F = number of students playing football = 37
• H = number of students playing hockey = 22
• |C ∩ F| = number of students playing at least cricket and football = 25
• |C ∩ H| = number of students playing at least cricket and hockey = 15
• |F ∩ H| = number of students playing at least football and hockey = 12
• |C ∩ F ∩ H| = number of students playing all three sports = 10

Step 2: Use the inclusion-exclusion principle

⇒ The formula for the number of students playing at least one of the sports is: |C ∪ F ∪ H| = |C| + |F| + |H| - |C ∩ F| - |C ∩ H| - |F ∩ H| + |C ∩ F ∩ H|

Step 3: Substitute the values

⇒  Substituting the values into the formula: |C ∪ F ∪ H| = 50 + 37 + 22 - 25 - 15 - 12 + 10

Calculating step by step:

• Sum of individual sports: 50 + 37 + 22 = 109
• Sum of pairwise intersections: 25 + 15 + 12 = 52
• Now substitute: |C ∪ F ∪ H| = 109 - 52 + 10 = 67

Step 4: Calculate the number of students not playing any sports

The total number of students is 100. Therefore, the number of students who do not play any of the three sports is: Students not playing any sport = 100 - |C ∪ F ∪ H| = 100 - 67 = 33

Thus, the number of students who do not play any of the three sports is: 1) 33