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Problem #33 · August 29, 2026

Natural numbers

How many natural numbers from 11 to 10001000 contain at least one digit 55 in their decimal representation?
  1. A271271
  2. B272272
  3. C290290
  4. D300300
  5. E352352

WORKED EXAMPLE

Solution

Count the complement: if we represent the numbers from 11 to 999999 as three-digit strings (001,…,999001, \ldots, 999), each position can contain any digit other than 55, giving 99 choices: 93=7299^3 = 729 strings, of which 000000 does not represent a number in this range — 728728 numbers without 55. 10001000 also contains no 55, giving 729729 in total. Thus, the count of numbers containing at least one 55 is 1000−729=2711000 - 729 = 271.
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