Question

Make the greatest and the smallest 4-digit number using any four different digits with the condition that the digit 9 is always at hundreds place.

Answer 1

Hint- Here, we will be choosing the different digits to be placed at different places in order to form the greatest and smallest 4-digit number respectively.
Given, digit 9 is always at hundreds 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 in hundreds place there is digit 9, so we are left with digits 0,1,2,3,4,5,6,7,8.
Also, digit 0 cannot be placed at thousands place because we need a 4-digit number not a 3-digit number.
For greatest 4-digit number,
The digit at thousands place should be greatest of 1,2,3,4,5,6,7,8. So, digit 8 should be there. Also, no digit can repeat. So, now we are left with digits 0,1,2,3,4,5,6,7. The digit at tens place should be greatest of 0,1,2,3,4,5,6,7. Hence, digit 7 will be placed at tens place and now we are left with 0,1,2,3,4,5,6. The digit at the ones place will be greatest of 0,1,2,3,4,5,6. Therefore, digit 6 will be placed at the ones place.
Hence, the greatest 4-digit number is 8976.
For smallest 4-digit number,
The digit at thousands place should be the smallest of 1,2,3,4,5,6,7,8. So, digit 1 should be there. Also, no digit can repeat. So, now we are left with digits 0,2,3,4,5,6,7,8. The digit at tens place should be the smallest of 0,2,3,4,5,6,7,8. Hence, digit 0 will be placed at tens place and now we are left with 2,3,4,5,6,7,8. The digit at the ones place will be the smallest of 2,3,4,5,6,7,8. Therefore, digit 2 will be placed at the ones place.
Hence, the smallest 4-digit number is 1902.

Note- These types of problems are solved simply by considering the possibilities of all the cases formed and 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.