Question

A thief broke into an Automated Teller Machine (ATM) using a screwdriver and was able to jam the card reader as well as breaking five keys from the keypad. The thief had to halt the process of break-in and hide, as a customer approached to use the ATM. The customer was able to successfully enter their ATM card, punch in the 4 digit PIN and was able to draw out some cash. Since the card reader was jammed, the customer was however not able to withdraw the ATM card, and drove off to seek some help. In the meantime, the thief came back and decided to try to discover the customer’s PIN so that he can steal money from the customer. You are required to calculate the maximum number of PINs that the thief may have to enter before correctly discovering the customer’s PIN?

Answer 1

the total number of keys are 10
(0,1,2,3,4,5,6,7,8,9).
But since the thief took out 5 keys, so only 5 keys are left.
So here n=5. (5keys left.)
and r=4 (4 digits are used for pin)
and since repetition is allowed.
Applying Permutation Formula,
${}^{n} P \: r = {}^{5} P \: 4 = {5}^{4} = 625$
max. no. of pins to be entered are 624
since 1 of the possible Permutation is the card holder pin itself