In a certain code language, the number "297" means "tie clip button". Which single digit stands for the word "button" in that code? Statement I: In that language, "926" means "clip your tie". Statement II: In that language, "175" means "hole and button".

Introduction / Context:
This data sufficiency question deals with a numerical code language. The three digit string "297" stands for three words "tie clip button". Further coded expressions and their translations are given. We must determine which single digit represents the word "button" and decide whether each statement alone, or both together, are sufficient to answer this uniquely.

Given Data / Assumptions:

• The code "297" corresponds to the set of words {tie, clip, button}.
• Statement I: "926" means "clip your tie".
• Statement II: "175" means "hole and button".
• Each English word maps to exactly one digit, and each digit maps to exactly one English word.
• Order of digits does not necessarily match word order.

Concept / Approach:
The standard approach to such code problems is to compare coded sequences that share some words, and then match common digits to common words. Intersection of code sets identifies which digit corresponds to each word. Crucially, we must check whether one statement alone already allows us to deduce the code for "button", or whether both are required, or whether the data remain insufficient.

Step-by-Step Solution:
Step 1: From the base information, the code set for "tie clip button" is {2, 9, 7} in some order. Step 2: Consider statement I: "926" means "clip your tie". The digit set {9, 2, 6} corresponds to the words {clip, your, tie}. Step 3: Compare the word sets {tie, clip, button} and {clip, your, tie}. The common words are "tie" and "clip". The corresponding common digits between {2, 9, 7} and {9, 2, 6} are 2 and 9. Step 4: Since 2 and 9 must correspond to "tie" and "clip", the remaining digit 7 must correspond to the remaining word, which is "button". Step 5: Therefore, statement I alone is sufficient to identify that the digit 7 stands for "button". Step 6: Now consider statement II by itself. "175" means "hole and button". The digit set {1, 7, 5} corresponds to {hole, and, button}. Step 7: The only word that appears both in this translation and in the base phrase "tie clip button" is "button". The only digit that appears both in the codes "297" and "175" is 7. Step 8: Therefore, statement II alone also allows us to conclude that 7 corresponds to "button". Step 9: Since each statement alone is sufficient, the correct data sufficiency category is that either statement I alone or statement II alone is sufficient to answer the question.

Verification / Alternative check:
Check with statement I: If 2 and 9 correspond to "tie" and "clip", and 7 corresponds to "button", then "926" representing "clip your tie" uses digits 9 and 2 for "clip" and "tie". The extra digit 6 must then represent "your", which is consistent. Check with statement II: With 7 as "button", the digits 1 and 5 represent "hole" and "and" in some order. There is no conflict with any mapping previously deduced. Thus both statements are individually consistent with the conclusion that 7 stands for "button".

Why Other Options Are Wrong:
Option a is incomplete because it ignores that statement II is also sufficient. Option b is similarly incomplete, ignoring the sufficiency of statement I. Option d is incorrect because both statements are not jointly required; each one by itself already gives the answer. Option e is wrong because the answer is clearly determined: the digit 7 stands for "button".

Common Pitfalls:
A common error is to think that both statements must always be used together, especially when two are provided. Another mistake is to overlook that the intersection method with only one additional sentence already isolates the code for "button". Some students also forget that the mapping from digits to words is one to one; once two of the digits are identified with two of the words, the remaining digit must map to the remaining word. Recognising these one to one correspondences is essential for quickly solving such questions.

Final Answer:
The digit 7 represents the word "button". Correct option: The data in either statement I alone or statement II alone are sufficient to answer the question.