Determine the number of natural numbers smaller than $10^4$, in the decimal notation of which all the digits are distinct.

Question

Determine the number of natural numbers smaller than $10^4$, in the decimal notation of which all the digits are distinct.

Please explain me the meaning of this question.

I am not asking a homework question. I just didn't understood meaning of this question. Please don't close my question

Answer 1

It means that that you must find all numbers in the range $1 \le n < 10^4$ without duplicate digits in their decimal notation. In some schools the range of "natural number" is $0 \le n < 10^4$.

$1234$

is allowed but not

$1134$

because it contains a repetition.

Answer 2

So first consider what we mean by a natural number in the decimal notation which is $\lt 10^4$. This is a positive integer with at most four digits.

We can write it as $$n=a_310^3+a_210^2+a_110^1+a_010^0=1000a_3+100a_2+10a_1+a_0$$ where we have $0\le a_0, a_1, a_2, a_3 \le 9$ and we normally write a decimal integer as a lot of digits with place value assumed $$n=a_3a_2a_1a_0$$ (where this is not multiplication but 'concatenation')

No repeated digits looks as though it means that $a_3, a_2, a_1, a_0$ are distinct when $n$ is expressed in this way. But some care is required with the digit zero, since we write $20$, which has no repeated digits, rather than $0020$, where the digits are repeated.