Question

From the greatest and smallest 4 – digit numbers using any four different digits with the condition that digit 5 is always at the ten’s place.

Answer 1

- Hint: In this question we will use the concept of place value and arrangement of numbers to form greatest and smallest numbers with given conditions. While making the greatest number, we will take the greatest digit in thousands place, next greatest digit in hundreds place, 5 in the tens place, and the third greatest digit in units place. Use a similar approach for the smallest number using the smallest digits in correct order.

Complete step-by-step solution -

We are to form a four digit greatest and smallest number where the tens place is fixed and digits do not repeat as all digits are to be different.
Now, in a 4 digit number, the highest place value is thousand. To form a greatest number, we will put the highest value digit in thousands place because. Greater the value of thousands place, greatest the number. We have a digit from 0-9, so will place the highest digit, that is 9 in thousands place.
Next hundred place, we will place the next highest digit possible, since 9 is already used, next highest will be 8.
So, in a hundred place we put 8. Tens place is already fixed by 5. So we will put the next highest digit which is not yet used in once place, which is 7.
Hence, the greatest 4 digit number with given condition will be 9857.
Now, let us form a smallest 4 digit number.
For smallest, we will put digits with least value in thousands places. The least digit is zero, but if we put zero in thousands place, then the resulting number will not be 4 digits. So, we will put the next least number, that is 1 in thousands place.
Now, for hundreds of places, the least possible digit is zero. So, we will put 0 in hundreds place.
Tens place is already fixed with 5. So, the next least number possible to place, that is 2, will be in unit place.
Hence, the smallest 4 digit number with required conditions is 1052.
Therefore, required numbers are 9857 and 1052.

Note: While forming the smallest 4-digit number, do not take 0 in the thousands place thinking that it is the smallest digit. If we place the digit 0 in a thousand place, then the number will be a 3- digit number. This will make the answer wrong. So, such mistakes should be avoided.