From $$50$$ students taking examinations in Mathematics, Physics and Chemistry, $$37$$ passed Mathematics, $$24$$ Physics and $$43$$ Chemistry. At most $$19$$ passed Mathematics and Physics, at most $$29$$ Mathematics and Chemistry and at most $$20$$ Physics and Chemistry. The largest possible number that could have passed all three exams is

Question

Infinity Learn · Accepted Answer

$$n(M$$∪P∪$$C)=50$$, $$n(M)=37$$, $$n(P)=24$$, $$n(C)=43$$, $$n(M$$∩P $$)$$≤$$19$$, $$n(M$$∩C $$)$$≤$$29$$, $$n(C$$∩P $$)$$≤$$20n(M$$∪P∪$$C)=n(M)+n(P)+n(C)-n(M$$∩P $$)-n(C$$∩P $$)$$−$$n(M$$∩$$C)+n(M$$∩P∩$$C)$$⇒ $$n(M$$∩P∩$$C)$$≤$$n(M$$∩$$P)+n(M$$∩$$C)+n(P$$∩$$C)$$−$$54$$ Therefore, the number of students that could have passed all three exams is at most $$19$$ $$+$$ $$29$$ $$+$$ $$20$$ $$-$$ $$54$$ $$=$$ $$14$$

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material