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Question

# How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?

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Solution

## Since the number is less than 1000, it means that it is a three-digit number, a two-digit number or a single-digit number. Three-digit numbers: The hundred's place can be filled by 5 digits neglecting zero as it can't be zero. The ten's place and the unit's place can be filled by 6 digits. So, total number of three digit numbers = 5$×$6$×$6 = 180 Two-digit numbers: The ten's place can be filled by 5 digits, except zero. The unit's digit can be filled by 6 digits. Total two digit numbers = 5$×$6 = 30 Single digit numbers are 1, 2, 3, 4, 5 as 0 is not a natural number. Thus, on neglecting it, we get 5 numbers. Total required numbers = 180 + 30 + 5 = 215

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