In a school of 100 students,70 studied English,30 studied French and 50 studied Latin.everyone studied atleast one of the subject,and 44 studied exactly two of the subject.how many students studied all the subject.
To find out how many students studied all three subjects, we can use the principle of inclusion-exclusion.
Let's denote:
A = the number of students who studied English
B = the number of students who studied French
C = the number of students who studied Latin
Given:
A = 70 (students who studied English)
B = 30 (students who studied French)
C = 50 (students who studied Latin)
A ∩ B = 44 (students who studied both English and French)
B ∩ C = ? (unknown)
A ∩ C = ? (unknown)
A ∩ B ∩ C = ? (unknown)
By applying the principle of inclusion-exclusion, we can calculate the number of students who studied all three subjects:
A ∪ B ∪ C = A + B + C - (A ∩ B) - (B ∩ C) - (A ∩ C) + (A ∩ B ∩ C)
100 = 70 + 30 + 50 - 44 - (B ∩ C) - (A ∩ C) + (A ∩ B ∩ C)
100 = 150 - 44 - (B ∩ C) - (A ∩ C) + (A ∩ B ∩ C)
(B ∩ C) + (A ∩ C) - (A ∩ B ∩ C) = 150 - 100 + 44
(B ∩ C) + (A ∩ C) - (A ∩ B ∩ C) = 94
Since everyone studied at least one subject, A ∪ B ∪ C = 100, so:
(B ∩ C) + (A ∩ C) - (A ∩ B ∩ C) = 100 - 44
(B ∩ C) + (A ∩ C) - (A ∩ B ∩ C) = 56
From the given information and calculations, we cannot determine the exact values of (B ∩ C), (A ∩ C), and (A ∩ B ∩ C). Therefore, we cannot determine the number of students who studied all three subjects.
To find out how many students studied all three subjects, we can use a method called the Principle of Inclusion-Exclusion. This principle states that the total number of elements in the union of two or more sets can be found by adding the sizes of the individual sets and subtracting the size of their intersection.
In this case, we have three subjects: English, French, and Latin. Let's denote the number of students who studied English as A, French as B, and Latin as C. We are given that A = 70, B = 30, and C = 50.
Now, we also know that everyone studied at least one subject, which means the total number of students is the union of these three sets, expressed as A ∪ B ∪ C. Using the Principle of Inclusion-Exclusion, we can write:
A ∪ B ∪ C = A + B + C - (A ∩ B) - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)
We are given that 44 students studied exactly two subjects, so we can substitute (A ∩ B), (A ∩ C), and (B ∩ C) as 44:
A ∪ B ∪ C = 70 + 30 + 50 - 44 - 44 - 44 + (A ∩ B ∩ C)
Simplifying further:
A ∪ B ∪ C = 62 + (A ∩ B ∩ C)
We want to find the value of (A ∩ B ∩ C), which represents the number of students who studied all three subjects. Rearranging the equation:
(A ∩ B ∩ C) = A ∪ B ∪ C - 62
Plugging in the values:
(A ∩ B ∩ C) = 62 + (A ∩ B ∩ C) - 62
Simplifying further:
0 = A ∩ B ∩ C
This means that the number of students who studied all three subjects is zero.