A four-digit PIN with no repeated digit

Question

A new phone forces you to pick a 4-digit PIN, and it chooses one for you completely at random, each of the digits can be anything from 0 to 9, and repeats are allowed. You glance at the result.

What is the probability that all four digits are different?

Accepted Answer

Count the total number of PINs first. Each of the four positions can independently be any of 10 digits, so there are 10 	imes 10 	imes 10 	imes 10 = 10{,}000 equally likely PINs. Now count the PINs with four distinct digits. Fill the positions left to right: the first digit is free (10 choices), the second must avoid the first (9 choices), the third must avoid both earlier digits (8 choices), and the fourth avoids all three (7 choices): 10 	imes 9 	imes 8 	imes 7 = 5040. So the probability is \frac{5040}{10{,}000} = \frac{63}{125} = 0.504. Just over half of random PINs use four differen…

A four-digit PIN with no repeated digit

Your answer

More Probability questions