In a class of 55 students, the number of students studying different subjects are, 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is _____.

Let M, P and C be the sets of students studying Mathematics, Physics and Chemistry respectively.

We have,

$n (M)=23, n (P) = 24, n (C) = 19, n (MP) = 12,$

$n(M∩C)=9, n(P∩C)=7$ and $n (M∩P∩C)=4$.

∴ Number of students studying exactly one subject

Using dot and crosses method

n(exactly one subject)

$=n (M) + n (P) + n (C) -2 \{n (M∩P) + n (P∩C)

+n (M∩C)\} + 3n (M∩P∩C)$

$=23+24+19-2(12+9+7) + 3× 4$

$=66-2 × 28 +12=22$