HOW TO FIND SUM OF ALL 4 DIGIT NUMBERS FORMED USING THE DIGITS

Here we are going to see how to find sum of all 4 digit numbers formed using the given digits 1, 2, 3, 4 and 5.

Let us look at some examples to understand the above concept.

Example 1 :

Find the sum of all 4-digit numbers that can be formed using digits 1, 2, 3, 4, and 5 repetitions not allowed ?

Solution :

Since we form a 4 digit number, let us create 4 places.

____    ____    ____    ____

Total number can be formed with the unit digit 1

 2  3  4  1  =  24

Total number can be formed starting with the digit 2

 2  3  4  1  =  24

Total number can be formed starting with the digit 3

 2  3  4  1  =  24

Total number can be formed starting with the digit 4

 2  3  4  1  =  24

Total number can be formed starting with the digit 5

 2  3  4  1  =  24

So, totally we may form 5(24)  =  120 numbers using the given digits.

From the above steps, we may understand that we may see the number 1 in 24 times in the unit place.

Sum of digits in the unit place 

  =  (1  24) + (2  24) + (3  24) + (4  24) + (5  24)

  =   360

Sum of digits in the ten's place 

  =  (1  24) + (2  24) + (3  24) + (4  24) + (5  24)

  =   360 ⋅ 10  =  3600

Sum of digits in the hundred's place 

  =  (1  24) + (2  24) + (3  24) + (4  24) + (5  24)

  =   360 ⋅ 100  =  36000

Sum of digits in the hundred's place 

  =  (1  24) + (2  24) + (3  24) + (4  24) + (5  24)

  =   360 ⋅ 1000  =  360000

Hence the sum of all 4 digit numbers formed with the given digits   

=   360000 + 36000 + 3600 + 360

=  399960

Example 2 :

Find the sum of all 4-digit numbers that can be formed using digits 0, 2, 5, 7, 8 without repetition?

Solution :

Sum of 4 digit numbers including 0

Since we form a 4 digit number, let us create 4 places.

___    ___    ___    ___

Total numbers can be formed with the unit digit 0 :

 2  3  4  1  =  24

Total number can be formed with the unit digit 2 :

 2  3  4  1  =  24

Total number can be formed with the unit digit 5 :

  2  3  4  1  =  24

Total number can be formed with the unit digit 7 :

  2  3  4  1  =  24

Total number can be formed with the unit digit 8 :

  2  3  4  1  =  24

Sum of digits in the unit place : 

  =  (0  24) + (2  24) + (5  24) + (7  24) + (8  24)

  =  0 + 48 + 120 + 168 + 144

  =  528

Sum of digits in the ten's place 

  =  528 ⋅ 10  =  5280

Sum of digits in the hundreds place 

  =  528 ⋅ 100  =  52800

Sum of digits in the thousand's  place 

  =  528 ⋅ 1000  =  528000

Hence the sum of 4 digit numbers including 0 

=  528000 + 52800 + 5280 + 528

  =  586608

Sum of 3 digit numbers excluding 0

Since we form a 4 digit number, let us create 4 places.

___   ____   ___  ____

Total number can be formed with the unit digit 2

 1  2  3  1  =  6

Total number can be formed with the unit digit 5

  1  2  3  1  =  6

Total number can be formed with the unit digit 7

 1  2  3  1  =  6

Total number can be formed with the unit digit 8

 1  2  3  1  =  6

Sum of digits in the unit place 

  =  (2  6) + (5  6) + (7  6) + (8  6)

  = 12 + 30 + 42 + 48

  =  132

Sum of digits in the ten's place 

  =  132 ⋅ 10  =  1320

Sum of digits in the hundreds place 

  =  132 ⋅ 100  =  13200

Hence the sum of 4 digit numbers excluding 0 

=  13200 + 1320 + 132

  =  14652

So, the sum of 4 digit numbers  =  586608 - 14652

  =  571956

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