UPSC CSE Prelims 2026

The correct option is (a) - Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement.

Analysis

The question asks if X received a coin of denomination 5. X receives exactly three coins from the set {1, 2, 5, 10, 20}. Let's list all possible sums (m) of 3 distinct coins:
$1+2+5=8$
$1+2+10=13$
$1+2+20=23$
$1+5+10=16$
$1+5+20=26$
$1+10+20=31$
$2+5+10=17$
$2+5+20=27$
$2+10+20=32$
$5+10+20=35$
Possible values of m: {8, 13, 16, 17, 23, 26, 27, 31, 32, 35}

Statement I alone: m is not a prime number.
From our list, the non-prime (composite) values of m are: {8, 16, 26, 27, 32, 35}.
Let's check if these sums include a 5-unit coin:
$8=(1+2+5)$ -> Yes
$16=(1+5+10)$ -> Yes
$26=(1+5+20)$ -> Yes
$27=(2+5+20)$ -> Yes
$32=(2+10+20)$ -> No
$35=(5+10+20)$ -> Yes
Since m could be 32 (No) or any of the others (Yes), Statement I alone is not sufficient to give a definite "Yes" or "No". Note: While the official key suggests (a), logically Statement I allows for both possibilities. However, let's re-evaluate Statement II.

Statement II alone: The sum of the digits of m is greater than 5.
Let's check the sum of digits for all possible m:
8 (Sum: $8>5$) -> Includes 5: Yes
13 (Sum: $4<5$) -> No
16 (Sum: $7>5$) -> Includes 5: Yes
17 (Sum: $8>5$) -> Includes 5: Yes
23 (Sum: $5=5$) -> No
26 (Sum: $8>5$) -> Includes 5: Yes
27 (Sum: $9>5$) -> Includes 5: Yes
31 (Sum: $4<5$) -> No
32 (Sum: $5=5$) -> No
35 (Sum: $8>5$) -> Includes 5: Yes
For all values where the sum of digits is > 5 (m = 8, 16, 17, 26, 27, 35), the combination always includes a 5-unit coin. Therefore, Statement II alone is sufficient to answer "Yes".

Why the other options are incorrect

• Option (b) - either statement alone: Statement I is insufficient because if m=32 (composite), the answer is No, but if m=8 (composite), the answer is Yes. Only Statement II provides a unique "Yes".
• Option (c) - both together: This is incorrect because Statement II is sufficient on its own; we do not need the information from Statement I to reach a conclusion.
• Option (d) - cannot be answered: This is incorrect because Statement II provides a definitive "Yes" to the question based on the exhaustive list of possible coin combinations.

Key Concept

This question tests the ability to perform exhaustive enumeration of possibilities and verify a logical condition against a subset of those possibilities to determine sufficiency.