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Question

# In a survey of 25 students, it was found that 12 have taken physics, 11 have taken chemistry and 15 have taken mathematics; 4 have taken physics and chemistry; 9 have taken physics and mathematics; 5 have taken chemistry and mathematics while 3 have taken all the three subjects. Find the number of students who have taken (i) physics only; (ii) chemistry only; (iii) mathematics only; (iv) physics and chemistry but not mathematics; (v) physics and mathematics but not chemistry; (vi) only one of the subjects; (vii) at least one of the three subjects; (viii) none of the three subjects.

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Solution

## Let P, C and Al be the sets of students who have taken physics, chemistry and mathematics respectively. Let a, b, c, d, e, f andg denote the number of students in the respective regions, as shown in the adjoining Venn. diagram. As per data given, we have ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩a+b+c+d=12b+c+e+f=11c+d+f+f+15 b+c=4 c+d=9 c+f=5 c=3 From these equations, we get c=3,f=2,d=6.b=1. Now, c+d+f+g=15⇒ 3+6+2+g=15⇒g=14;b+c+e+f=11⇒ 1+3+e+2=11⇒e=5;a+b+c+d=12⇒a+1+3+6=12⇒a=2;∴ a=2,b=1,c=3,d=6,e=5,f=2 and g=4 So, we have: (i) Number of students who offered physics only = a= 2. (ii) Number of students who offered chemistry only = e= 5. (iii) Number of students who offered mathematics only g = 4. (iv) Number of students who offered physics and chemistry but not mathematics = b =1. (v) Number of students who offered physics and mathematics but not chemistry = d = 6. (vi) Number of students who offered only one of the given subjects = (a+e+g) = (2 +5+4) = 11. (vii) Number of students who offered at least one of the given subjects=(a+b+c+d+c+f+g)=(2 +1 +3 +6+5 +2+4)= 23. (viii) Number of students who offered none of the three given subjects = (25 - 23) = 2.

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