Exercise 185Combinatorics › Counting principlesHigh schoolAdded 31 Oct 2026

Four-digit numbers with distinct digits

#counting#arrangements

Statement

How many four-digit numbers (from $1000$ to $9999$ ) have four different digits?

Hint

Choose the digits one at a time, starting with the first.

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The first digit can be any of $1,\dots,9$ : $9$ choices (it cannot be $0$ ).
The second digit can be any digit except the first, including $0$ : $9$ choices.
The third digit: $8$ choices.
The fourth digit: $7$ choices.

By the multiplication principle, there are $9\cdot9\cdot8\cdot7=4536$ such numbers.