In a certain code language, the word TROUPE is written as GILFKV and the word ERUPTION is written as VIFKGRLM, using a fixed substitution for each letter. Using the same substitution pattern, how is the word GENIUS written in that code language?

Introduction / Context:
Here we have a pure substitution cipher: every letter of the alphabet is consistently replaced by another letter. Two words and their codes are given, TROUPE → GILFKV and ERUPTION → VIFKGRLM. We must determine how the word GENIUS will look under the same substitution.

Given Data / Assumptions:

• TROUPE → GILFKV.
• ERUPTION → VIFKGRLM.
• The substitution is one-to-one and consistent across all examples.
• We must encode GENIUS.

Concept / Approach:
We first match each letter of TROUPE with its coded form, then do the same for ERUPTION, and see which letter–letter mappings repeat. This allows us to build a mapping dictionary from original letters to coded letters. Then we simply apply this dictionary to the letters of GENIUS.

Step-by-Step Solution:
Step 1: From TROUPE → GILFKV: T → G, R → I, O → L, U → F, P → K, E → V. Step 2: From ERUPTION → VIFKGRLM: E → V, R → I, U → F, P → K, T → G, I → R, O → L, N → M. Step 3: Combine both to get the mapping: E → V, R → I, U → F, P → K, T → G, O → L, I → R, N → M, G → T (from consistency with options), and S → H (from consistency with options). Step 4: Now encode GENIUS. G → T, E → V, N → M, I → R, U → F, S → H. Step 5: Putting them together gives TVMRFH.

Verification / Alternative check:
This mapping fits both given words perfectly without any conflict. Among the options, only TVMRFH matches the pattern ?VMRF? expected from the partial mapping T, V, M, R, F, ?. The last letter can only be H if S is mapped to H, which keeps the cipher one-to-one.

Why Other Options Are Wrong:
Other options break the established mapping. For example, some options change the encoded value of U or E or introduce letters that would clash with already used cipher letters. Therefore they cannot be correct under a consistent substitution cipher.

Common Pitfalls:
Learners sometimes assume that letter shifts are arithmetic (like +2 or -3 across the entire word), but here each letter has its own specific replacement. Another mistake is to ignore consistency across both given examples; valid mappings must satisfy both words simultaneously.

Final Answer:
Using the same substitution, GENIUS is written as TVMRFH in that code language.