In the cinema set of a movie, 125 mechanical aliens were created. Some of these aliens had peculiar features. 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?

### Concept & Formula To find the number of elements with *none* of the features, we first calculate the total number of elements that have *at least one* feature using the standard 3-set Union formula, and then subtract that from the universal total. $$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$$ ### Step-by-Step Solution * **Given:** * Universal Total = $125$ * Noses ($N$) = $40$, Legs ($L$) = $30$, Ears ($E$) = $20$ * $N \cap L$ = $10$, $L \cap E$ = $12$, $N \cap E$ = $5$ * $N \cap L \cap E$ = $3$ * **Calculation:** * First, plug the given values directly into the union formula to find how many aliens have at least one unusual feature: * Total with features = $40 + 30 + 20 - 10 - 12 - 5 + 3$ * Total with features = $90 - 27 + 3 = 66$ * Now, subtract the aliens with features from the total created aliens to find those without any: * None = Total - (Total with features) * None = $125 - 66 = 59$ * Since $59$ is not present in options (a), (b), or (c), the answer is "None of these". ### Exam Strategy & Shortcut For standard formula applications, group all the positive terms and negative terms separately for faster mental math: $(40+30+20+3) - (10+12+5) = 93 - 27 = 66$. Then compute $125 - 66 = 59$. Avoid drawing Venn diagrams when the data strictly maps to the union formula components. ### Common Pitfall A frequent error is assuming that "$10$ had two noses and three legs" means *only* those two features. In set theory, unless the word "only" is used, the intersection includes those who might have the third feature as well. The standard formula handles this overlap perfectly. ### Final Answer **Therefore, the correct answer is None of these.**