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 ?

Solution

Total number of digits = 6  Clearly, the natural numbers ten's than 1000 can be 3 digits, 2 digits and 1 digit numbers. Now, 0 cannot be a first digit of the three digit numbers.  So, the hundred's place can be filled with a.ny of the 5 digits I, 2, 3....5. So, there are 5 ways of filling the hundred place.  The ten's place can be filled with in any of the 6 digits 0, 1, 2 .... 5. So, there are 6 ways of filling the ten's place.  The unit's place can be filled with in any of the 6 digits 0, 1, 2.... 5. So, there are 6 ways of filling the ten's place. The total number of 3 digit numbers = 5×6×6=180 Similarly, the total number of 2 digit numbers = 5×6=30 Now, 0 is not a natural number  ∴ the total number of 1 digit numbers = 5  ∴ Total number of natural numbers tens than 1000 = 180 + 30 + 5 = 215  MathematicsRD SharmaStandard XI

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