Question

Make the greatest and the smallest 4-digit numbers by using any one digit twice:
$\left( i \right)$ 6, 3, 2.
$\left( {ii} \right)$ 1, 0, 6.

Answer 1

Hint: Use the concept that the greatest number is starting from the highest number followed by least numbers and smallest number is starting from least number followed by highest numbers so use this concept to reach the solution of the question.

Complete step-by-step answer:

We have to make the smallest and greatest 4-digit numbers by using any one digit twice.
$\left( i \right)$ 6, 3, 2.
Now as we see there are 3 digits given and we have to consider the fourth digit among these digits.
So first we consider 6 as a fourth digit.
So the digits become 6, 6, 3 and 2.
Now we have to make a smallest and greatest four digit number using these digits.
Smallest number – As we know the smallest number is starting from lowest number followed by greatest number so among these numbers smallest number is 2 second smallest number is 3 and there are two six so third and fourth place is filled by 6.
Therefore the smallest number is (2366).
Greatest number - As we know greatest number is starting from largest number followed by smaller number so among these numbers largest number is 6 and second largest number is also 6 as there are two six, third largest number is 3 and fourth largest number is 2 respectively.
Therefore the largest number is (6632).
Now we consider 3 as a fourth digit.
So the digits become 6, 3, 3 and 2.
Now we have to make a smallest and greatest four digit number using these digits.
Smallest number – As we know smallest number is starting from lowest number followed by greatest number so among these numbers smallest number is 2 second smallest number is 3 and third place is also filled by 3 as there are two three and fourth place is filled by largest number which is 6.
Therefore the smallest number is (2336).
Greatest number - As we know greatest number is starting from largest number followed by smaller number so among these numbers largest number is 6 and second largest number is 3 and third place is also filled by 3 as there are two 3 and fourth place is filled by least digit which is 2.
Therefore the largest number is (6332).
Now we consider 2 as a fourth digit.
So the digits become 6, 3, 2 and 2.
Now we have to make a smallest and greatest four digit number using these digits.
Smallest number – As we know smallest number is starting from lowest number followed by greatest number so among these numbers smallest number is 2 second smallest number is also 2 and third place is filled by 3 and fourth place is filled by largest number which is 6.
Therefore the smallest number is (2236).
Greatest number - As we know greatest number is starting from largest number followed by smaller number so among these numbers largest number is 6 and second largest number is 3 and third place is also filled by 2 and fourth place is also filled by 2 as there are two 2.
Therefore the largest number is (6322).
$\left( {ii} \right)$ 1, 0, 6.
Now as we see there are 3 digits given and we have to consider the fourth digit among these digits.
So first we consider 6 as a fourth digit.
So the digits become 6, 6, 0 and 1.
Now we have to make a smallest and greatest four digit number using these digits.
So as we know that we cannot start the number from 0 because if we start the number from 0 the number converts into a three digit number for example (0123 = 123) is a three digit number.
Smallest number – As we know the smallest number is starting from the lowest number followed by the greatest number but in this case it is not starting from 0 it is starting from 1 then 0 then 6 and then again 6.
Therefore the smallest number is (1066).
Greatest number - As we know greatest number is starting from largest number followed by smaller number so among these numbers largest number is 6 then again 6 then 1 then 0.
Therefore the largest number is (6610).
Now we consider 1 as a fourth digit.
So the digits become 6, 0, 1 and 1.
Now we have to make a smallest and greatest four digit number using these digits.
Smallest number – As we know the smallest number is starting from the lowest number followed by the greatest number but in this case it is not starting from 0 it is starting from 1 then 0 then 1 and then 6.
Therefore the smallest number is (1016).
Greatest number - As we know greatest number is starting from largest number followed by smaller number so among these numbers largest number is 6 then 1 then again 1 then 0.
Therefore the largest number is (6110).
Now we consider 0 as a fourth digit.
So the digits become 6, 0, 0 and 1.
Now we have to make a smallest and greatest four digit number using these digits.
Smallest number – As we know the smallest number is starting from the lowest number followed by the greatest number but in this case it is not starting from 0 it is starting from 1 then 0 then again 0 and then 6.
Therefore the smallest number is (1006).
Greatest number - As we know greatest number is starting from largest number followed by smaller number so among these numbers largest number is 6 then 1 then 0 then again 0.
Therefore the largest number is (6100).

Note: In such types of question 0 digit in given digits and we have to use all the digits then in the case of smallest number we cannot start the number from 0 because the number converts into a three digit number for example (0123 = 123) is not a four digit number it is a three digit number then apply the concept that the greatest number is starting from highest number followed by least numbers and smallest number is starting from least number followed by highest numbers, we will get the required answer.