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In a class of 37 students, 25 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football ?
5
6
3
4
4
Just add the two groups (25 + 16 = 41) and subtract the total number of students (41 - 37 = 4). Wait, checking values: 25+16=41, 41-37=4. Re-evaluating: The sum of sets exceeds total by the number of students counted twice (both).
Total number of students n(U) = 37, students who like Cricket n(C) = 25, students who like Football n(F) = 16. Every student likes at least one game, so n(C \cup F) = 37.
n(CтИкF)=n(C)+n(F)тИТn(CтИйF)
Just add the two groups (25 + 16 = 41) and subtract the total number of students (41 - 37 = 4). Wait, checking values: 25+16=41, 41-37=4. Re-evaluating: The sum of sets exceeds total by the number of students counted twice (both).
Students often confuse the union (total class size) with the individual sets and forget to subtract the total population from the sum of the two subsets.
Define set theory parameters
Let n(C) be students who like cricket and n(F) be students who like football. Given n(C)=25, n(F)=16, and the union n(CтИкF)=37.
n(C)=25,n(F)=16,n(CтИкF)=37
Apply inclusion-exclusion principle
Use the formula n(CтИкF)=n(C)+n(F)тИТn(CтИйF) to find the intersection, where n(CтИйF) represents students who like both.
n(CтИйF)=n(C)+n(F)тИТn(CтИкF)
Calculate the final result
Substitute the values: n(CтИйF)=25+16тИТ37. This equals 41тИТ37=4.
n(CтИйF)=4
D is correct because the calculation 25+16тИТ37 results in 4, indicating 4 students like both games.
This concept of overlapping sets is fundamental for solving questions on probability distributions and data interpretation in competitive exams.