Question

Make the greatest and the smallest 4–digit number using any four different digits with conditions as given.
Digit 4 is always at tens place

Answer 1

Hint: Here we take the numbers less than 10 as we have to find the smallest and greatest 4 digit numbers. Keep the 4 at ten’s place. Try to determine the number which will be in thousand, hundred and unit’s place to make the number greatest and smallest.

Complete step-by-step answer:
Given, Digit 4 is always at tens place
Since, we have to form a 4-digit number so there will be digits at its ones, tens, hundreds and thousands place

The different digits which can be used to make any number are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
We already know that at tens place there is digit 4 so, we are left with digits 0, 1, 2, 3, 5, 6, 7, 8, and 9
Also digit zero cannot be places at thousand because we need 4-digit number

For Greater 4-digit number,
The digit at thousands place should be greatest of 1, 2, 3, 4, 5, 6, 7, 8, 9
So digit 9 should be there
Also no digit can repeat, so now we are left with digits 1 , 2, 3, 4, 5, 6, 7, 8
The digit at hundreds place should be greatest of 1, 2, 3, 4, 5, 6, 7, 8 Hence digit 8 will be placed at hundreds place

Now, we are left with 1, 2, 3, 4, 5, 6, 7. The digit at ones place should be greatest of 1, 2, 3, 4, 5, 6, 7
Therefore digit 7 will be place at ones place
Hence the greatest four digit number is 9847.


For Smallest 4-digit number,
The digit at thousands place should be smallest of 1, 2, 3, 4, 5, 6, 7, 8, 9
So digit 1 should be there
Here digit 0 is not used at thousand places as it will convert the number into three digit number
Also no digit can repeat, so now we are left with digits 0, 2, 3, 4, 5, 6, 7, 8, 9
The digit at hundreds place should be smallest of 0, 2, 3, 4, 5, 6, 7, 8, 9 Hence digit 0 will be placed at hundreds place
Now, we are left with 2, 3, 4, 5, 6, 7, 8, 9. The digit at ones place should be smallest of 2, 3, 4, 5, 6, 7, 8, 9
Therefore digit 2 will be place at ones place
Hence the smallest four digit number is 1042.

Note: These types of problems are solved simply by considering the possibility of all the cases formed afterwards going with the one that is asked for. Here we placed the most appropriate digit at different places in order to make the greatest and smallest 4-digit number provided the given condition also holds. And be careful whether we have to consider zero or not.