Question

Write the smallest $7$-digit number using four different digits.

Answer 1

Here, in the given question, we are asked to find the number which should be of seven digits and also it should be the smallest seven digit number using four digits. To find the required number, we will first find the smallest seven digit number. And then add the numbers to get the required number.

Complete step-by-step solution:
Smallest $n$-digit can be found by putting $1$ at the first place and $0$ at the remaining i.e. $n - 1$ places.
Hence, the smallest seven digit number will be $10,00,000$. But this number has only two different digits i.e.$0,1$, we will add $1$ in the unit place. Then the number becomes $10,00,001$ which again has two different digits only $0,1$. If we move further to check the numbers starting from $10,00,002$ to $10,00,009$, we will see that these numbers contain only three different digits. Moving further, $10,00,010$ and $10,00,011$ again have only two different digits. Similarly, if we add $1$ to each number till we find the required number, we will find that the numbers starting from $10,00,012$ to $10,00,022$ are having three different digits only. Now, adding $1$ again we get, $10,00,023$ which is the seven digit number having four different digits.
Hence, the smallest seven-digit number having four different digits is $10,00,023$.

Note: We have a total of ten numbers available i.e. $0,1,2,3,4,5,6,7,8,9$. We will choose the four smallest numbers out of the total numbers available because we are asked to find the smallest seven digits number using four different digits. And we know that $0,1,2,3$ are the four smallest numbers out of all. It means we have to arrange these four digits only to get the desired result.