Question

How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when repetition of digits is allowed?

Answer 1

Hint: For 3 digit numbers, we can choose the first digit any of 5 ways. (but can't be 0).
Choose the second digit any of 6 ways. Similarly, for the third digit, we can choose any of 6 ways. Total number of 3 digits natural numbers are \[5\times 6\times 6=180\] . For 2 digit numbers, we can choose the first digit any of 5 ways. (but can't be 0). Then, the second digit can be any of 6 ways. Total number of 3 digits natural numbers are \[5\times 6=30\] . For 1-digit numbers, we can choose the first digit any of 5 ways.

Complete step-by-step answer:
We have to find the number of natural numbers that are less than 1000. It means the number can be 3 digit natural number, 2 digit natural number and also 1 digit natural number.
Now, for a three digit natural number, at 1st digit place we have 5 digits and we can’t include 0 because that will make our number as 2 digit number. But we can include 0 as possible digits for 2nd and 3rd digit places. So we have 6 possible digits. Let us understand with a diagram.

The total number of 3 digit numbers \[=5\times 6\times 6=180\] .
Now, for a two digit natural number, at 1st digit place we have 5 digits and we can’t include 0 because that will make our number as 1 digit number. But we can include 0 as possible digits for the 2nd digit place. So we have 6 possible digits. Let us understand with a diagram.

The total number of 3 digit numbers \[=5\times 6=30\] .
Now, for 1 digit natural number, at 1st place we have 5 digits and we can’t include 0 because zero is not a natural number.
Total number of natural numbers \[=180+30+5=215\] .
Hence, the total number of natural numbers is 215.

Note:This question can also be solved in another way.
We need to find the natural number which is less than 1000. So, in the first place we can take any digit from 0 to 5. It means we have 6 possible digits for the 1st digit place. Similarly, we have 6 possible digits for the 2nd digit place and 3rd digit place.
Total numbers \[=6\times 6\times 6=216\] .
But this also includes 000, but 0 is not a natural number. So, we deduct this number.
Total number of natural numbers \[=216-1=215\] .