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; (iui) 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; (vii) none of the three subjects.

Answer

(i) `=5`.
(ii) `=4` .
(iii) `=2`.
(iv) `=1`.
(v) `=6`.
(vi) `=11`.
(vii) `=23`.
(viii) `=2`.