found that 15 take mathematics and 12 has taken physics 11 has taken chemistry 5 has taken chemistry and mathematics and chemistry 9 has taken mathematics
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(i) 11−5−4+3=5
(3 removed twice)
(ii) 4−3=1
(iii) only maths =15−9−5+3
=4
only phy =12−9−4+3=2
only chem =5⇒Total=11
(iv) Atleast one =11+9+5+4−2(3)
=29−6=23
⇒ None =25−23=2
(v) 23
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