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In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most?
Most Upvoted Answer
In a class of 95 students, each student plays at least one of hockey, ...
Given information:
- Total number of students in the class = 95
- Number of students playing hockey = 55
- Number of students playing cricket = 60
- Number of students playing football = 65
To find:
- The maximum number of students who play all three games
Solution:
Step 1: Find the number of students who play at least two games
- Let's assume the number of students who play hockey and cricket is 'x'.
- Let's assume the number of students who play cricket and football is 'y'.
- Let's assume the number of students who play hockey and football is 'z'.
Step 2: Find the number of students who play exactly two games
- Number of students who play exactly two games = x + y + z
Step 3: Find the number of students who play all three games
- Number of students who play all three games = x + y + z
Step 4: Use the principle of inclusion-exclusion to find the maximum possible value of x, y, and z
- Total number of students who play at least two games = x + y + z
- Total number of students who play hockey = 55
- Total number of students who play cricket = 60
- Total number of students who play football = 65
Using the principle of inclusion-exclusion:
- Total number of students who play at least two games = Total number of students who play hockey + Total number of students who play cricket + Total number of students who play football - Number of students who play exactly two games - 2 * Number of students who play all three games
Step 5: Calculate the maximum number of students who play all three games
- Total number of students who play at least two games = x + y + z = (55 + 60 + 65) - Number of students who play exactly two games - 2 * Number of students who play all three games
- Since each student plays at least one of the games, the minimum value of x, y, and z is 0.
- Therefore, the maximum number of students who play all three games can be calculated as follows:
- Total number of students who play at least two games = (55 + 60 + 65) - Number of students who play exactly two games - 2 * Number of students who play all three games
- Maximum number of students who play all three games = (55 + 60 + 65) - Number of students who play exactly two games - 2 * Number of students who play all three games
Step 6: Calculate the maximum number of students who play exactly two games
- The maximum number of students who play exactly two games can be calculated by subtracting the maximum number of students who play all three games from the total number of students who play at least two games.
- Maximum number of students who play exactly two games = Total number of students who play at least two games - Maximum number of students who play all three games
Step 7: Calculate the maximum number of students who play all three games
- Plug in the values and calculate the maximum number of students who play all three games.
Final
- Total number of students in the class = 95
- Number of students playing hockey = 55
- Number of students playing cricket = 60
- Number of students playing football = 65
To find:
- The maximum number of students who play all three games
Solution:
Step 1: Find the number of students who play at least two games
- Let's assume the number of students who play hockey and cricket is 'x'.
- Let's assume the number of students who play cricket and football is 'y'.
- Let's assume the number of students who play hockey and football is 'z'.
Step 2: Find the number of students who play exactly two games
- Number of students who play exactly two games = x + y + z
Step 3: Find the number of students who play all three games
- Number of students who play all three games = x + y + z
Step 4: Use the principle of inclusion-exclusion to find the maximum possible value of x, y, and z
- Total number of students who play at least two games = x + y + z
- Total number of students who play hockey = 55
- Total number of students who play cricket = 60
- Total number of students who play football = 65
Using the principle of inclusion-exclusion:
- Total number of students who play at least two games = Total number of students who play hockey + Total number of students who play cricket + Total number of students who play football - Number of students who play exactly two games - 2 * Number of students who play all three games
Step 5: Calculate the maximum number of students who play all three games
- Total number of students who play at least two games = x + y + z = (55 + 60 + 65) - Number of students who play exactly two games - 2 * Number of students who play all three games
- Since each student plays at least one of the games, the minimum value of x, y, and z is 0.
- Therefore, the maximum number of students who play all three games can be calculated as follows:
- Total number of students who play at least two games = (55 + 60 + 65) - Number of students who play exactly two games - 2 * Number of students who play all three games
- Maximum number of students who play all three games = (55 + 60 + 65) - Number of students who play exactly two games - 2 * Number of students who play all three games
Step 6: Calculate the maximum number of students who play exactly two games
- The maximum number of students who play exactly two games can be calculated by subtracting the maximum number of students who play all three games from the total number of students who play at least two games.
- Maximum number of students who play exactly two games = Total number of students who play at least two games - Maximum number of students who play all three games
Step 7: Calculate the maximum number of students who play all three games
- Plug in the values and calculate the maximum number of students who play all three games.
Final
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In a class of 95 students, each student plays at least one of hockey, ...
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In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most?
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In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most?.
In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most?.
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