There are 175 students in a class, out of which 100 students opt Maths, 70 opt Physics and, 23 opt Physics and Chemisty, 28 opt Chemistry and Maths and 18 opt of all three subjects. Now find each of the following :
No. of students who opt why only Maths
(ii) No. of students who opt only Physics
(iii) No. of students who opt only Chemistry
(iv) No. of students who opt Physics and Chemisty but not Maths
(v) No. of students who opt Maths and Physics but not Chemistry
(vi) No. of students who opt only one subject
(viii) No. of students who opt atleast one subject
(viii) No. of student who do not opt any subject.

Explanation

Let A = set of students opted Maths
B = set of student opted Physics
C = set of students opted Chemistry
and U = set of students of the class
Given that `n(U)=175,n(A)=100,n(B)=70,n(C)46`
`n(A capB)=30,n(BcapC)=23,n(C cap A)=28,n(A cap B cap C)=18`
(i) No. of students opted Maths only
`=n(A capB'capC')`
`=n[A cap(B cupC)']`
`=n(A)-n[A cap(B cupC)]`
`[becausen(AcapB')=n(A)-n(AcapB)]`
`=n(A)-n[(AcapB)cup(A capC)]`
`= n(A)-[n(AcapB)+n(AcapC)-n(A capBcapC)]`
`=100-30-28+18=60`.
(ii) No. of students opted Physics only
`= n(BcapA'capC')`
`=n(B)-[n(BcapC)+n(BcapA)-n(A capBcapC)]`
`=70 -(23+30 - 18)=35`.
(iii) No. of students opted Chemistry only
`n(CcapA'capB')`
`=n(C)-[n(CcapA)+n(CcapB)-n(A capBcapC)]`
`=46-(28+23 - 18)=13`.
(iv) No. of students opted Physics and Chemistry but not Maths
`= n(BcapCcapA')`
`=n(B capC)-n(BcapCcapA)`
`=23-18=5`.
(v) No. fo student opted Maths and Physics but not Chemistry
`n(AcapBcapC')=n(A capB)-n(A capBcapC)`
`=30-18=12`
(vi) No. of students taken only one subject
`=n(A)+n(B)+n(C)-2n(A capB)-2n(BcapC)-2n(C capA)+3n(A capBcapC)`
`= 100 + 70 + 46 - 2 (30 + 23 + 28) + 3 xx 18`
`= 216 - 162 + 54 = 108`.
(vii) No. of students taken atleast one subject
`n(AcupBcupC)=n(A)+n(B)+n(C)-n(AcapB)-n(BcapC)-n(CcapA)+n(AcapBcapC)`
`=100 +70+46 - (30+23 +28)+18`
`=234 - 81 = 153`
(viii) No. of students taken no subject `= n(U) - n (A cup B cup C) = 175 - 153 = 22`
Second method : Here M = set of students taken Maths
P = set of students taken Physics
C = set of students taken Chemistry
Consider the Venn diagram

`n(M)=a+b+d+e=100`
`n(P)=b+c+e+f=70`
`n(C)=d+e+f+g = 46`
`n(M cap P)=b+e = 30`
`n(P cap C)=e +f = 23`
`n (M cap C)=d+e = 28`
`n (M cap P cap C) = r = 18`
`n(U) = a+b+c+d+e+f+g+h`
`= 175`
On solving
`e=18`
`d+18=28rArrd=10`
`18+f=23rArrf=5`
`b+18=30 rArr b = 12`
`n(C) = d+e + f + g = 46`
`rArr 10 + 18 + 5 + g = 46`
`rArr g = 13`
`n (P) = b+c+e+f = 70`
`rArr 12 + c + 18 + 5 = 70`
`rArr c = 35`
`n(M) = a+b+d+e = 100`
`rArr a+12 +10+18 = 100`
`rArr a = 60`
and `n(U)=a+b+c+d+e+f+g+h`
`= 175`
`rArr 60 + 12 + 35 + 10+ 18 + 5 + 13 + h = 175`
`rArr h = 22`
Now, (i) No. of students taken Maths only `= a= 60`.
(ii) No. of students taken Physics only = c = 35
(iii) No. of student taken Chemisty only = g = 13.
(iv) No. of students taken Physics and Chemistry but not Maths = f = 5
(v) No. of students taken Maths and Physics but not Chemistry = b = 12
(vi) No. of students taken only one subject
`= a + c +g`
`= 60 + 35 + 13 = 108`
(vii) No, of students taken atleast one subject
`= a+b+c+d+e+f+g`
`= 60 + 12 + 35+ 10 + 18 + 5 + 13`
`= 153`
(viii) No. of student taken no subject
`= h = 22`.