 # How many natural numbers less than 1000 can be formed from the digits

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:

To find: number of natural numbers less than 1000 that can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times

For forming a 3 digit number less than 1000 possible ways are:

$5 \times 6 \times 6 \ldots$ (in 100 's place 5 digits are only possible 0 not included.) $=180$

For forming a 2 digit number less than 1000 possible ways are:

$5 \times 6 \ldots \ldots$ (in 10 's place 5 digits are only possible 0 not included.)

$=30$

For forming a 1 digit number less than 1000 possible ways are:

$5 \ldots(0$ not included because it is a whole number and natural number is asked in question.)

So total number of numbers less than 1000 that can be formed from the digits $0,1,2,3$, 4,5 when a digit may be repeated any number of times $=180+30+5=215$