+1 vote
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Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?

1. 0
2. 2
3. 4
4. 8

edited | 394 views
Migrated from GO Mechanical 11 months ago by Arjun

Total Students who watched Film $= 40$ Let

• $A =$ Students who watched Film "A" alone
• $B =$ Students who watched Film "B" alone
• $C =$ Students who watched Film "C" alone
• $X =$ Students who watched Both Film "A" and "B" but not Film "C"
• $Y =$ Students who watched Both Film "A" and "C" but not Film "B"
• $Z =$ Students who watched Both Film "B" and "C" but not Film "A"
• $O =$ Students who watched all the 3 Films

By Principle of Inclusion and Exclusion, we have

Total Students who watched Film $= A + B + C + X + Y + Z + O$

Now from the Venn diagram it is clear that $13 = A + X + Y + O$

No student watched exactly $2$ films. Hence, $X = Y = Z = 0$

Therefore, $13 = A + 0 + 0 + O$

$\implies 13 - O = A$

Similarly, $B = 16 - O$ and $C = 19 - O$

So, Total Students who watched Film $= A + B + C + X + Y + Z + O$

$\implies 40 = A + B + C + 0 + 0 + 0 + O$

$\implies 40 = A + B + C + O$

$\implies 40 = (13 - O) + (16 - O) + (19 - O) + O$

$\implies 40 = 48 - 2O$

$\implies -8 = -2O$

$\implies O = 4.$

Hence, C is correct.

by Loyal
edited
+1
X = Students who watched Both Film "A" and "B"

should be

X = Students who watched Both Film "A" and "B" but not Film "C"
+1
+1 vote

$|AUBUC|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|$

$40=13+16+19-x-x-x+x$

$2x=8$

$x=4$

$Note: Why\ |A\cap B|,|B\cap C|,|A\cap C|\ is\ x\ ?$

Each student watched either only one film or all three which means $A\ \&\ B\ but\ not\ C=0$