Directions for the next 5 (five) items : Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly. Question : X receives three coins of different denominations : 1, 2, 5, 10 and 20. If the total amount received by X is m, does X receive a coin of denomination 5? Statement I : m is not a prime number. Statement II : The sum of the digits of m is greater than 5.

The correct answer is Option A. To determine if X receives a coin of denomination 5, we must evaluate the possible combinations of exactly three different coins chosen from the set {1, 2, 5, 10, 20}. Using the combinations formula (5C3), there are exactly 10 possible sets.

First, let us identify the sums (m) of the 4 combinations that do not contain the 5-rupee coin:

• 1 + 2 + 10 = 13 (Sum of digits: 1+3 = 4)
• 1 + 2 + 20 = 23 (Sum of digits: 2+3 = 5)
• 1 + 10 + 20 = 31 (Sum of digits: 3+1 = 4)
• 2 + 10 + 20 = 32 (Sum of digits: 3+2 = 5)

The remaining 6 sets that do contain the 5-rupee coin yield sums (m) of 8, 16, 17, 26, 27, and 35.

Evaluating Statement I: 'm is not a prime number.' The non-prime sums across all combinations are 8, 16, 26, 27, 32, and 35. This list includes m=32 (which does not contain a 5-rupee coin) and m=8 (which does). Since it yields mixed results, Statement I alone is mathematically insufficient.

Evaluating Statement II: 'The sum of the digits of m is greater than 5.' As calculated above, the combinations lacking a 5-rupee coin have digit sums of 4, 5, 4, and 5. None of these are greater than 5. Consequently, if the sum of the digits is strictly greater than 5, m must correspond to a combination that includes the 5-rupee coin (whose digit sums are 8, 7, 8, 8, 9, and 8). Statement II alone gives a definitive 'Yes' and is entirely sufficient.

Why other options are wrong:

• Option B is incorrect because both statements are not independently sufficient; Statement I fails to give a definitive answer.
• Option C is incorrect because the statements do not need to be combined; Statement II alone is enough.
• Option D is incorrect because the question can indeed be answered using Statement II.

Takeaway: In CSAT Data Sufficiency logic puzzles, comprehensively listing the inverse scenarios (e.g., all sums without the 5 coin) is the fastest way to test conditional boundaries.