Question

Find out the greatest and the smallest 4-digit numbers using any four different digits with conditions as given:
Digit 1 is always at thousands’ place.

Answer 1

Hint: Fix the digit 1 at the thousands’ place. Then in the case of finding the smallest 4-digit number using four different digits, arrange from the lower digits to the higher, without using 1, in the hundreds’, tens’ and ones place respectively in ascending order. Again, in the case of finding the greatest 4-digit number using four different digits, arrange from the higher digits to the lower, without using 1, in the hundreds’, tens’ and ones place respectively in descending order.

Complete step-by-step answer:
For greatest 4-digit number with four different digits with 1 in thousands’ place, we arrange the digits in descending order as follows:
$\begin{align}
  & \text{Thousands Hundreds Tens Ones} \\
 & \text{ 1 9 8 7} \\
\end{align}$
Thus, the greatest 4-digit number with four different digits with 1 in thousands’ place is 1987.
For smallest 4-digit number with four different digits with 1 in thousands’ place, we arrange the digits in ascending order as follows:
$\begin{align}
  & \text{Thousands Hundreds Tens Ones} \\
 & \text{ 1 0 2 3} \\
\end{align}$
Thus, the greatest 4-digit number with four different digits with 1 in thousands’ place is 1023.

Note: In case of the smallest 4-digit number, we don’t use 1 in the tens’ place after using 0 in the hundreds’ place, as we are being asked to use 4 different digits to make the smallest number. Also, we need to start from the lowest digit 0 on the hundreds’ place to get the lowest number. Similarly, we need to start from the highest digit 9 on the hundreds’ place to get the highest number.