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Can numbers such as 7710 and 0137 be allowed in this condition? No, as 7710 is formed with
repetition of digits, and 0137 is not a 4-digit number. Here, 7310 is the greatest number and 1037
is the smallest number that are formed with the digits 1, 0, 3, 7 only once.

Thus, numbers can be formed using the given digits in two ways—without repetition of digits and
with repetition of digits.
To form numbers using the given digits without repetition, proceed as follows:

(a) To form the smallest number with the given digits, write the smallest digit other than zero (0)
in the leftmost place, then write zero if given, otherwise write other digits in their ascending
order.
(b) To form the greatest number with the given digits, write the greatest digit in the leftmost place,
then write other digits in their descending order.

Note: If 0 is one of the given digits, then we put 0 at the second place from the extreme left position in order to write
the smallest number by using the given digits.

Example 4: Form:
(a) The smallest number using the digits: 6, 4, 5, 7, 2, 3, 0.

(b) The largest number using the digits: 1, 4, 8, 5, 2, 7.
Solution: (a) The smallest number using the digits 6, 4, 5, 7, 2, 3, 0 is 2034567.
(b) The largest number using the digits 1, 4, 8, 5, 2, 7 is 875421.

Example 5: Make the greatest and the smallest 5-digit numbers using different digits in which 5
appears in the tens place.

Solution: The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and it is given that 5 appears in the tens place.
For greatest number, place 9 at the highest place and then 8 and 7, and put 6 in units place.

Therefore, the greatest number is 98756.
Now, for the smallest number, take the digit 1 at the highest place, as a number cannot start with
0. After that, use 0 and 2 in the next places and put 3 in units place.
Therefore, the smallest number is 10253.

To form numbers using given digits with repetition, proceed as follow:

(a) To form the smallest number, repeat the smallest digit (other than 0) to the extreme left
positions and the remaining digits in ascending order.
(b) To form the greatest number, repeat the greatest digit in the extreme left positions and the
remaining digits in descending order.

Note: Since, ‘0’ is never written on the left most place. In case ‘0’ is to be repeated, then it is repeated just after the
first place from the leftmost place for the smallest number and in the last place for the largest number.

Example 6: Form the smallest and greatest 8-digit number using the digits 6, 7, 3, 2 and 1.
Solution: The given digits are 6, 7, 3, 2 and 1.

Smallest number: 11112367

Greatest number: 77776321

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