In a certain number code, the word MISTAKE is written as 9765412 and the word NAKED is written as 84123. Using the same digit code, how will the word DISTANT be written?

Introduction / Context:
This question uses a letter-to-digit coding system. The words MISTAKE and NAKED are given with their numeric codes, and you must use this mapping to encode a new word, DISTANT. Because all the letters in DISTANT appear among the letters of MISTAKE and NAKED, we can derive the complete mapping needed. Such questions test your ability to construct a digit mapping and apply it accurately to a new word without confusion.

Given Data / Assumptions:

• MISTAKE → 9765412.
• NAKED → 84123.
• All letter–digit mappings are consistent across words.
• Target word: DISTANT.
• No rearrangement; we simply write digits corresponding to each letter in order.

Concept / Approach:
We first assign digits to letters using the code for MISTAKE. Then we refine and extend this mapping by using the code for NAKED. Once the mapping is complete, we encode DISTANT by replacing each letter with its corresponding digit. This is a direct substitution problem: each letter is replaced by a fixed digit wherever it appears.

Step-by-Step Solution:
Step 1: From MISTAKE → 9765412, map each letter position-wise: M → 9, I → 7, S → 6, T → 5, A → 4, K → 1, E → 2. Step 2: From NAKED → 84123, add new mappings and cross-check existing ones: N → 8, A → 4 (same as before, consistent), K → 1 (consistent), E → 2 (consistent), D → 3. Step 3: The letter set for DISTANT is D, I, S, T, A, N, T. Step 4: Substitute each letter: D → 3 I → 7 S → 6 T → 5 A → 4 N → 8 T → 5 Step 5: Combine digits in order: 3 7 6 5 4 8 5 → 3765485.

Verification / Alternative check:
To verify, you can check that this mapping reconstructs the original words. Using the mapping M → 9, I → 7, S → 6, T → 5, A → 4, K → 1, E → 2, MISTAKE indeed becomes 9765412. Similarly, N → 8, A → 4, K → 1, E → 2, D → 3 turns NAKED into 84123. Furthermore, if you decode 3765485 using the same mapping, you get DISTANT correctly. This consistency in forward and backward direction ensures that the mapping is correct and that 3765485 is the valid code for the word DISTANT.

Why Other Options Are Wrong:
Options B, C, D, and E all change at least one digit in a way that would correspond to changing the mapped letter. For example, any option that places a digit other than 3 for D, or other than 8 for N, or changes the repeated digit 5 for T, breaks the mapping established from the given words. Once we fix the mapping from MISTAKE and NAKED, there is only one possible sequence of digits for DISTANT, which is 3765485, making the other options impossible.

Common Pitfalls:
A common error is to misread or miscopy digits from the given words or to assume that letter positions might also affect digits. In this question, the mapping is purely letter-based and position-independent. Candidates also occasionally reverse the order of digits or forget that repeated letters must always correspond to the same digit. Building a clear mapping table and double-checking each substitution helps avoid these mistakes.

Final Answer:
Hence, in the given number code, the word DISTANT is written as 3765485.