Finding properties of the sum of all permutations of 3 distinct digits
Question
Let A, B and C represent distinct non-zero digits. Suppose x is the sum of all possible 3-digit numbers formed by A, B and C without repetition. Consider the following statements:
Which of the above statements is/are correct?
Options
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Explanation
For any 3 distinct digits, the sum x of all their 6 possible permutations is given by the formula: x = 222 × (A + B + C).
Statement 1 asks for the least possible value. To minimize x, pick the smallest distinct non-zero digits: \{1, 2, 3\}. x = 222 × (1 + 2 + 3) = 222 × 6 = 1332. Since 1332 is a 4-digit number, Statement 1 is precisely correct.
Statement 2 asks for the greatest 3-digit value. However, we just proved that the absolute mathematical minimum for x is 1332. Because the lowest possible value is already 4 digits long, ⟨MATH⟩x⟨/MATH⟩ can never be a 3-digit number under any circumstances. Statement 2 is entirely false.
Answer: (a).
Question details
Year
2022
Paper
CSAT
Question
Q54
Section
Numerical Ability
Sub-topic
Number Puzzles
Type
Statement-based
Difficulty
Hard
Source hint
Sum of all possible 3-digit numbers formed by A, B, C
Same sub-topic — other years
Number Puzzles has appeared in multiple CSAT papers:
See all questions on Number Puzzles
Browse every tagged question across all years