In a class, 22 offer maths, 21 offer English and 25 offer science. 4 students offer all three subjects and 36 offer exactly one subject. If all the students offer at least one of the subject, then how many offer two subjects?
Suppose we let
a = math only
b = English only
c = science only
x = math and science only
y = math and English only
z = English and science only
If you draw the Venn diagram, you see that
a+b+c = 36
a+x+y+4 = 22
b+y+z+4=21
c+x+z+4=25
Solve those equations and you get
x = b-7
y = c-11
z = 28-b-c
x+y+z is the number with exactly two subjects
x+y+z = b-7+c-11+28-b-c = 10
To find out how many students offer two subjects, we can use the principle of inclusion-exclusion.
We know that 4 students offer all three subjects, and 36 students offer exactly one subject. Let's denote the number of students offering only maths as M, the number of students offering only English as E, and the number of students offering only science as S.
From the given information, we can write the following equations:
M + E + S = 36 (Equation 1) (number of students offering exactly one subject)
M + E + S + 4 = Total number of students (Equation 2) (all students offer at least one subject)
Now, let's solve these equations simultaneously.
Subtracting Equation 1 from Equation 2, we get:
(M + E + S + 4) - (M + E + S) = Total number of students - 36
4 = Total number of students - 36
Total number of students = 4 + 36
Total number of students = 40
Now, to find the number of students offering two subjects, we need to find the values of M, E, and S.
Substituting the value of Total number of students in Equation 2, we get:
M + E + S + 4 = 40
M + E + S = 36
From Equation 1, we have:
36 = 36
This means that the values of M, E, and S are all equal to 12. So, we can conclude that 12 students offer only maths, 12 students offer only English, and 12 students offer only science.
Therefore, the number of students offering two subjects is:
Total number of students - (students offering only one subject) - (students offering all three subjects)
40 - 36 - 4 = 40 - 40 = 0
Hence, none of the students offer two subjects.