Data Sufficiency – Code Language (Three-Digit Words) Question: In a code language, '297' means 'tie clip button'. Which single digit represents 'button'? Statements: I. '926' means 'clip your tie'. II. '175' means 'hole and button'.

Introduction / Context: Another code-language Data Sufficiency item. We know '297' encodes three distinct words: tie, clip, button. We must decide if Statement I or II (or both) lets us isolate which single digit stands for 'button'.

Given Data / Assumptions:

• Base: '297' → {tie, clip, button}.
• I: '926' → {clip, your, tie}.
• II: '175' → {hole, and, button}.
• One-to-one mapping within the problem’s language.

Concept / Approach: Use set intersections. If two coded strings share exactly the words 'clip' and 'tie', then the shared digits correspond to those two words; the remaining digit in '297' must then be 'button'. Similarly, if you compare a string containing 'button' with the base, the unique common digit reveals the code for 'button'.

Step-by-Step Solution (using I only): '297' → {2,9,7} for {tie, clip, button}. '926' → {9,2,6} for {clip, your, tie}. Common words between base and I: {clip, tie} → common digits {9,2}. Therefore the remaining digit in '297', which is 7, must represent 'button'.

Alternative (using II only): '175' encodes {hole, and, button}. Intersect digits with '297': {1,7,5} ∩ {2,9,7} = {7}. Since both code groups contain the word 'button', the unique common digit 7 must be 'button'.

Why Other Options Are Wrong: 'I alone sufficient while II alone not' and vice versa are both too restrictive because each statement, independently, already pins down 'button'. 'Both I and II' is stronger than necessary. 'Neither' is incorrect because we actually can deduce the digit.

Common Pitfalls: Forgetting that the base information '297' → 'tie clip button' is always available when testing each statement’s sufficiency.

Final Answer: Either I or II is sufficient