Question

One Hundred Twenty-five $\left(125\right)$ aliens descended on a set of film as Extra Terrestrial Beings. $40$ had two noses, $30$ had three legs, $20$ had four ears, $10$ had two noses and three legs, $12$ had three legs and four ears, $5$ had two noses and four ears and $3$ had all the three unusual features. How many were there without any of these unusual features?

• A. $5$
• B. $35$
• C. $80$
• D. $59$

Answer 1

The correct option is D $59$

n(Two noses) $=n\left(N\right)$

n(Three Legs) $=n\left(L\right)$

n(Four Ears) $=n\left(E\right)$

$n\left(U\right)=125$

$n\left(N\right)=40,n\left(L\right)=30,n\left(E\right)=20$

$n(N\cap L)=10,n(L\cap E)=12,n(N\cap E)=5,n(N\cap E\cap L)=3$

$n(N\cup L\cup E)=n\left(N\right)+n\left(L\right)+n\left(E\right)-n(N\cap L)-n(N\cap E)-n(L\cap E)+n(N\cap E\cap L)$

$n(N\cup L\cup E)=40+30+20-5-10-12+3=66$

Aliens without these three unusual features $=n\left(U\right)-n(N\cup L\cup E)$

$=125-66=59$