A 4-digit PIN is made by arranging four of the digits 0 to 9. If the same digit may be used any number of times and the first digit may be 0, which of the following is the number of possible PINs?

Answer: C. 10,000

Each of the four positions can take any of the 10 digits 0 to 9, so the number of permutations with repetition is 10⁴ = 10,000. If the same digit could not be reused, there would be 10P4 = 10 × 9 × 8 × 7 = 5,040, and if in addition the order were ignored, 10C4 = 210. This number is also a measure of the effort of a brute-force attack that tries every PIN in turn.

• A. This is the number of combinations 10C4, without repeated digits and without regard to order. In a PIN, the order of the digits matters.
• B. This is the number of permutations without repeated digits, 10P4 = 10 × 9 × 8 × 7. The question allows the same digit to be used any number of times.
• C. Correct. Each position has 10 choices, so 10⁴ = 10,000.