Counting valid PIN combinations based on numerical constraints
Question
There is a numeric lock which has a 3-digit PIN. The PIN contains digits 1 to 7. There is no repetition of digits. The digits in the PIN from left to right are in decreasing order. Any two digits in the PIN differ by at least 2. How many maximum attempts does one need to find out the PIN with certainty?
Options
6
8
10
12
Explanation
Let the PIN be d_1 d_2 d_3. The rules are: d_1 > d_2 > d_3, with d_1 - d_2 \ge 2 and d_2 - d_3 \ge 2. Consequently, d_1 - d_3 must be at least 4. Valid digits are \{1, 2, 3, 4, 5, 6, 7\}. Systematically list candidates bounding by the smallest digit d_3:
\rightarrow d_2 = 3 \implies d_1 \in \{5, 6, 7\} (3 combos) \rightarrow d_2 = 4 \implies d_1 \in \{6, 7\} (2 combos) \rightarrow d_2 = 5 \implies d_1 = 7 (1 combo)
\rightarrow d_2 = 4 \implies d_1 \in \{6, 7\} (2 combos) \rightarrow d_2 = 5 \implies d_1 = 7 (1 combo)
\rightarrow d_2 = 5 \implies d_1 = 7 (1 combo)
Summing all valid permutations: 3 + 2 + 1 + 2 + 1 + 1 = 10 attempts.
Answer: (c).
Question details
Year
2022
Paper
CSAT
Question
Q55
Section
Logical & Analytical Reasoning
Sub-topic
Permutations & Combinations
Type
Factual single
Difficulty
Medium
Source hint
Numeric lock 3-digit PIN, constraints
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