A PIN With a Repeat

Question

A 4-digit PIN is any string of four digits from 0 to 9 (leading zeros allowed), so there are 10^4 = 10000 in total. How many PINs contain at least one repeated digit, meaning some digit appears two or more times?

Accepted Answer

Complement count. The opposite of "some digit repeats" is "all four digits distinct," which is a simple sequential count: 10 choices, then 9, then 8, then 7: 10 cdot 9 cdot 8 cdot 7 = 5040. Subtract from the total: 10^4 - 5040 = mathbf4960. So just under half of all PINs are "all distinct," a number worth remembering as a sanity anchor for birthday-problem questions. Approach 2, direct case count (the hard way) Split by repeat pattern: exactly one pair (AABC): 10 cdot binom42 cdot 9 cdot 8 = 4320; two pairs (AABB): binom102 cdot frac4!2!,2! = 270; a triple (AAAB): 10 cdot binom43 cdot 9 = 360; all four equal: 10. Sum: 4320 + 270 + 360 + 10 = 4960. ✓ The trap: attempting the direct route first and botching one of the four patterns (the two-pair term is the usual casualty, its frac4!2!2! arrangement factor gets replaced by binom42 or forgotten). The complement is one line and immune to case-slicing errors. The reflex "at least one = total minus none" is the single highest-value habit in counting, and it is exactly the move behind the birthday paradox: the all-distinct count here is the same product that drives collision probabilities.

A PIN With a Repeat

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