Question

You just purchased the lockbox below to store a spare set of keys for your house. It allows you to set a passcode using digits from 0 to 9:

How it works:

-Each digit is either ON or OFF, so pushing the same button twice removes it from the combination (as opposed to adding it a second time to the code). E.g. pressing 2, 7, 4 or pressing 2, 7, 3, 4, 3 would produce the same code: 274

-It is a mechanical lock box (not digital) so it does not recognize the order of the buttons pushed. E.g. pressing 4, 5, 8, 1 is the same as pressing 8, 1, 4, 5: both represent the same code (4581 = 8145)

-Pressing the CLEAR button clears the last code attempted (sets all digits to OFF)

a) Explain why the correct code cannot exceed 10 digits.

b) There is no manual that comes with the lockbox. Your friend thinks the code needs to be exactly 10-digits long. If this were true, explain why this would not be a good lock box.

c) What is the probability of randomly guessing the correct code on your first attempt? Explain how you arrived at your formula.

d) Another friend suggests using 9-digit codes over 2- or 3-digit codes as they are longer and better for security. Do you agree or disagree? Explain.

Answer 1