distinctdigit

Distinctdigit refers to integers whose decimal representation uses no repeated digits. In base 10, a number is distinct-digit if each digit from 0 to 9 appears at most once. The digit 0 may appear, but it cannot be the leading digit.

Examples include 7, 28, 1023456789, and 9876543210. Numbers such as 110 or 122345 are not distinct-digit because they contain repeated digits.

Counting distinct-digit numbers in base 10: for an n-digit number with distinct digits (1 ≤ n ≤ 10), the first digit has 9 choices (1–9), and each subsequent digit has decreasing choices from the remaining digits, including 0. Thus the count is 9 × 9 × 8 × 7 × … up to n factors. For example, n = 1 gives 9 numbers; n = 2 gives 81; n = 3 gives 648. The total number of distinct-digit numbers with up to 10 digits is 9 × 9! = 3,265,920. The largest 10-digit distinct-digit number is 9876543210.

In programming and puzzles, a common task is to test whether a given integer has distinct digits. A typical approach uses a bitmask or a boolean array to track digits seen so far, scanning digits with division and modulus operations. If a digit repeats, the number is not distinct-digit. This method runs in time proportional to the number of digits.

Generalizations extend the idea to other bases. In base b, an n-digit distinct-digit number (n ≤ b) has (b−1) × P(b−1, n−1) possibilities, where P denotes a permutation count. The concept thus encompasses a family of digit-distinctness problems across numeral systems.