Repeated Digit in a Random Code

It is easier to compute the complementary probability that all five digits are distinct.

Count the ways to pick five distinct digits in order, out of all $10^5$ codes.

1. 1. Count codes with all distinct digits

   There are $10\times 9\times 8\times 7\times 6$ codes with no repeated digit.

   $$10\times 9\times 8\times 7\times 6=30240$$

2. 2. Subtract from one

   Divide by the total $10^5$ codes and take the complement.

   $$P=1-\frac{30240}{100000}=\frac{436}{625}$$

Answer: 436/625