Making numbers: Numbers can be made using the given digits with or without repetition of digits.
The possible digits are \(0, 1, 2 , 3, 4, 5, 6, 7, 8\) and \(9\).
1. To form the smallest number using the given digit the smallest digit should be kept at the leftmost and the remaining digits in ascending order of their values to the right.
2. To form the largest number using the given digit the largest digit should be kept at the leftmost and the remaining digits in descending order of their values to the right.
Here come two cases for the formation of a number using its digits:
- Without repeating digits
- With repeating digits
Without repetition: In this type, we have to use the given digits only once and form the number.
Make the greatest and smallest three-digit number using the digits \(2, 5, 7\).
Now let us write all the possible number with the digits.
The possible numbers are \(275, 257, 572, 527, 725,\) and \(752\).
It is clear from the above numbers that the greatest number is \(752\) and the least number is \(257\).
We can directly form the greatest and least number by arranging its digits in ascending and descending from left to right.
With repetition: In this type, we can use the given digits as per the repetition rule provided in the problem and form the number.
Make the greatest and the smallest 4-digit numbers by using any one digit twice. The digits are \(8, 0, 5\).
Arranging the number in ascending order becomes \(0, 5, 8\).
Arranging the number in descending order becomes \(8, 5, 0\).
Here zero is the least digit. But if we use zero in the leftmost place, we will get only the two-digit number \((0058)\).
So let us take the next smallest digit as leftmost digit and then place \(0\) to get the smallest four-digit number.
Since one of the digits repeats twice, let us make \(0\) repeats to get the least number.
That is the smallest four-digit number is \(5008\).
For largest four-digit number, let us repeat the number \(8\) twice.
That is the largest four-digit number is \(8850\).
If zero is one of the digits while forming the smaller number, it should be kept after the next highest number. Because if we place zero in the leftmost, the number will be lost its place value.
For example, \(0234\) is a three digit-number. But \(2034\) is a four-digit number with the same digits in both the case.