Question

The sum of all the four digit even numbers which can be
formed by using the digits 0, 1, 2, 3, 4 and 5 if repetition of
digits is allowed​

Answer 1

Answer: We will have an even number when the units digit is either 0, 2 or 4

Also, 1,000th

can be only 1, 2, 3 or 4 (NOT 0) as we want a 4 digit number

Case 1: Unit digit is 0

1,000th

digit has 4 choices (1, 2, 3, 4) and sum of these digits =10

100th

digit has 5 choices (0, 1, 2, 3, 4) and sum of these digits =10

10th

digit has 5 choices (0, 1, 2, 3, 4) and sum of these digits =10

Unit digit has 1 choices (0) and sum of these digits =0

Now, each of the 1,000th

digit will feature in 25 numbers

Similarly, each of 100th

and 10th

digit will feature in 20 numbers

Also, each of the Unit digit will feature in 100 numbers

Hence, Sum of these numbers =10⋅25⋅1000+10⋅20⋅100+10⋅20⋅10+0⋅100⋅1=250,000+20,000+2,000+0=272,000

Case 2: Unit digit is 2

1,000th

digit has 4 choices (1, 2, 3, 4) and sum of these digits =10

100th

digit has 5 choices (0, 1, 2, 3, 4) and sum of these digits =10

10th

digit has 5 choices (0, 1, 2, 3, 4) and sum of these digits =10

Unit digit has 1 choices (2) and sum of these digits =2

Now, each of the 1,000th

digit will feature in 25 numbers

Similarly, each of 100th

and 10th

digit will feature in 20 numbers

Also, each of the Unit digit will feature in 100 numbers

Hence, Sum of these numbers =10⋅25⋅1000+10⋅20⋅100+10⋅20⋅10+2⋅100⋅1=250,000+20,000+2,000+200=272,200

Case 3: Unit digit is 4

1,000th

digit has 4 choices (1, 2, 3, 4) and sum of these digits =10

100th

digit has 5 choices (0, 1, 2, 3, 4) and sum of these digits =10

10th

digit has 5 choices (0, 1, 2, 3, 4) and sum of these digits =10

Unit digit has 1 choices (4) and sum of these digits =4

Now, each of the 1,000th

digit will feature in 25 numbers

Similarly, each of 100th

and 10th

digit will feature in 20 numbers

Also, each of the Unit digit will feature in 100 numbers

Hence, Sum of these numbers =10⋅25⋅1000+10⋅20⋅100+10⋅20⋅10+4⋅100⋅1=250,000+20,000+2,000+400=272,400

Total Sum of All Numbers =250,000+272,200+272,400=794,600

hope this will help you