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In a survey of 400 students, 200 study Physics, 250 study Chemistry, and 100 study both. How many students study neither?
50
100
150
200
50
Use the formula: Total = (P + C - Both) + Neither. Calculate (200 + 250 - 100) = 350, then subtract 350 from 400 to get 50.
Total number of students n(U) = 400, students studying Physics n(P) = 200, students studying Chemistry n(C) = 250, students studying both n(P тИй C) = 100.
n(PтИкC)=n(P)+n(C)тИТn(PтИйC)
Use the formula: Total = (P + C - Both) + Neither. Calculate (200 + 250 - 100) = 350, then subtract 350 from 400 to get 50.
Students often add 200 and 250 and subtract that sum from 400, forgetting that the 100 students studying both are counted twice in the sum of 450.
Calculate Union of Sets
Find the total number of students studying at least one subject using the Inclusion-Exclusion principle: n(PтИкC)=n(P)+n(C)тИТn(PтИйC).
n(PтИкC)=200+250тИТ100=350
Calculate Students Studying Neither
Subtract the number of students who study at least one subject from the total number of students in the survey.
n(Neither)=n(U)тИТn(PтИкC)=400тИТ350=50
A is correct because 350 students study either Physics or Chemistry, leaving 50 students who study neither.
This concept is a precursor to Probability theory, specifically for calculating P(AтИкB)=P(A)+P(B)тИТP(AтИйB).