**A number of students prepared for an examination in physics, chemistry and mathematics. Out of this number, 15 took physics, 20 took chemistryand 23 took mathematics, 9 students took both chemistry and mathematics, 6 student took both physics and mathematics and all those who took physics also took chemistry. One student fell ill and failed to write any of the papers. I. Represent the information on a Venn diagram II. How many students took all three subjects? III. How many students took exactly one of the subject? IV. How many students took exactly two of the subjects? V. How many students prepared for the examination?**

The **Answer to the Question**

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**Here's the Solution to this Question**

I.

II.

$N(P\cap C\cap M)=6$

6 students took all three subjects.

iii.

$N(P\cap C')=0=>N(P\cap C'\cap M')=0$

$N(C\cap P'\cap M')=20-(15+9-6)=2$

$N(M\cap P'\cap C')=23-9=14$

$14+2=16$

16 students took exactly one of the subjects.

IV.

$N(P\cap C\cap M')=15$

$N(P\cap M\cap C')=0$

$N(M\cap C\cap P')=9-6=3$

$15+3=18$

18 students took exactly two of the subjects.

V.

$0+2+14+9+3+0+6=34$

$34+1=35$

35 prepared for the examination.