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?

The PIN contains three digits out of – 1, 2, 3, 4, 5, 6, and 7

Case I: The rightmost digit is 1
The possible combinations are: 531, 631, 731, 641, 741, 751 (i.e. 6 possible combinations)

Case II: The rightmost digit is 2
The possible combinations are: 642, 742, 752 (i.e. 3 possible combinations)

Case III: The rightmost digit is 3
The possible combinations are: 753 (i.e. 1 possible combination)

The rightmost digit cannot be more than 3. (due to difference of 2 between each)

So, the total number of possible combinations of the PIN = 6 + 3 + 1 = 10