The door lock at your new laboratory requires a passcode. The $5-$digit code consists of all of

the digits $1$ through $5,$ which must be entered in a secret sequence, without repetition. $($For example,

$21345$ could be a valid code, but $435$ would not be valid, since it is only $3$ digits, and $14225$ would not

be valid, since it contains repetition and doesn't contain a $3.)$

Arriving at the laboratory door, you realize you forgot to bring the paper on which the lab supervisor

wrote down the code. But...

you know that paper is on your desk in your dormitory, which is a $15-$minute round trip bike ride

from the lab;

you remember that the rst digit of the code is $4,$ but don't remember any of the rest of the

sequence;

you know it takes about $3$ seconds to enter a passcode, and the keypad requires a $27-$second wait

after entering an incorrect code so each wrong attempt would take about $30$ seconds total; and

you know a building alarm will sound, notifying campus police, if the correct code is not entered

within $15$ minutes after an incorrect code entry.

Which of the following two options is a more ecient use of your time?

Option $($A$)$ Ride back to your dorm, retrieve the paper with the passcode, then return to the lab in

$15$ minutes and enter the code correctly on the rst try.

Option $($B$)$ Systematically enter each possible code beginning with a $4,$ one at a time, until you get

lucky and the lab door opens.

$($Do not suggest other alternatives: no one will hear if you pound on the door, and you don't have

contact information to ask any other sta for the passcode.$)$

Support your answer with calculations that show the amount of time required to guarantee success for

option B$.$