Question

How many natural numbers are there from $1$ to $1000$ which have none of their digits repeated?

Answer 1

Hint: We have 3 kinds of digits from $1$ and $1000$ such as one digit numbers, two digit numbers and three digit numbers.

Complete step by step answer:

Now here we have to find natural numbers where none of the digits should be repeated.
Here from $1$ to $1000$ we have $3$ kinds of digits
One digit numbers, two digit numbers and three digit numbers.

One digit numbers:
We know that there are $9$ possible to get single digit numbers from $1-9$
$\Rightarrow 9ways$
Two digit numbers:
Here the first digit can be from $1-9$ and the second digit can be from$0-9$.
We also know that “zero” cannot be the first digit so we have excluded it
Total possible = $9\times 9=81$ ways
Three digit numbers:
Here the first digit can be from $1-9$ and the second digit can be from $0-9$ but not the first digit $10-1=9$.
And the third digit can be from $0-9$ but not the same as the first and second digit.
Total possible=$9\times 9\times 8=648$

Here we have found all the possible under without repetition condition
Therefore total number of natural numbers from $1$ to $1000$ without repetition= $648+81+9=738$ ways.

Note: Make a note that digits should not be repeated and kindly focus that zero can’t be the first digit for any kind terms.