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Senior Manager  P
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What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 56% (02:10) correct 44% (02:14) wrong based on 184 sessions

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What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A. 103324

B. 93324

C. 87645

D. 77678

E. 65432

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Originally posted by fitzpratik on 29 Sep 2018, 20:31.
Last edited by Bunuel on 30 Sep 2018, 04:13, edited 1 time in total.
Renamed the topic and edited the question.
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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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4
1
fitzpratik wrote:
What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A. 103324

B. 93324

C. 87645

D. 77678

E. 65432

let us take 2 as the thousands digit
the following combinations are possible
2345
2435
2354
2534
2453
2543

let us take sum of units digit
2*5 + 2*4 + 2*3 = 10+8+6 = 24

similarly
when we take 3 as the thousands digit
2*5 + 2*2 + 2*4 = 10+4+8=22

when we take 4 as the thousands digit
2*5 + 2*2 + 2*3 = 10+4+6=20

when we take 5 as the thousands digit
2*2 + 2*3 + 2*4 = 4+6+8=18

sum of units digit = 84

similarly sum of tens digit = 84*10
sum of hundreds digit =84*100
sum of thousands digit = 84*1000

sum total = 84+84*10+84*100+84*1000 = 84(1111) = 93324
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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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The way I solved it was using the formula suggested by bunuel

on the following link:
https://gmatclub.com/forum/what-is-the- ... 94836.html
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What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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2
I solved this question using another approach,
Since there are 4! Or 24 Ways of arranging the digits without repetition, the sum of all the numbers formed should be a multiple of 24. Using POE I got OA.Although I got OA Bunuel is this reasoning applicable to questions of this sort? Thanks.

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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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fitzpratik wrote:
What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A. 103324

B. 93324

C. 87645

D. 77678

E. 65432

Dear Experts,
Kindly help with a solution on this one please. Thank you.
VP  P
Joined: 07 Dec 2014
Posts: 1224
Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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3
2
fitzpratik wrote:
What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A. 103324

B. 93324

C. 87645

D. 77678

E. 65432

add least and greatest: 2345+5432=7777
7777/2=3888.5=average
4! possible combinations=24
24*3888.5=93324
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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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Four digit numbers =4⋅3⋅2⋅1=24 ways we can form a four digit number. Since it's a 4 digit number, each digit will appear 6=24/4 times in each of units, tens, hundreds, and thousands place. Therefore, the sum of digits in the units place is 6(2+3+4+5)=85

Why they are adding 2+3+4+5?= Because each digit will come at those places.

Similarly sum of digits at tens,hundreds,thousands place is 84

Required sum of all numbers =84+84⋅10+84⋅100+84⋅1000=93324

poojasingavi wrote:
fitzpratik wrote:
What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A. 103324

B. 93324

C. 87645

D. 77678

E. 65432

Dear Experts,
Kindly help with a solution on this one please. Thank you.

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Posts: 9704
Location: Pune, India
Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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poojasingavi wrote:
fitzpratik wrote:
What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A. 103324

B. 93324

C. 87645

D. 77678

E. 65432

Dear Experts,
Kindly help with a solution on this one please. Thank you.

Request to all forum users:

I am one of the forum experts on the PS and CR forums. If you would like me to take a look at any question on these forums, please mention it on my thread (link given below):

PS - https://gmatclub.com/forum/veritas-prep ... 78042.html
CR - https://gmatclub.com/forum/veritas-prep ... 78026.html

Since I receive 100s of mails everyday in my mailbox, unfortunately, it becomes very difficult to follow up on every query even if I am tagged on it. I do take a look at both my threads on a daily basis and hence will certainly get to your query if you mention on it.

Thank you!

Coming to the question:

What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A few things to note here:
The numbers will be of the form 2354, 3425 ... etc
There will be a total of 4*3*2*1 = 24 numbers
These will be made symmetrically. What this means is that in the 24 numbers, 2 will appear 6 times, 3 will appear 6 times, 4 will appear 6 times and 5 will appear 6 times (since all digits are equivalent)

2345
3245
...
2354
3254
...

etc

This will be true of digits in tens place, hundreds place and thousands place too.

The sum of digits in units place = 6*2 + 6*3 + 6*4 + 6*5 = 6*(2 + 3 + 4 + 5) = 84
The sum of digits in tens places, hundreds places and thousands places will also be 84 each.

The sum of all numbers then will be = 84 + 84*10 + 84*100 + 84*1000 = 93324

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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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fitzpratik wrote:
What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?

A. 103324

B. 93324

C. 87645

D. 77678

E. 65432

GMATinsight : sir is there any direct formula to solve such questions?
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GMAT 1: 620 Q43 V33 GMAT 2: 690 Q47 V38 Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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Hi,

If we are not told can we assume that the numbers can repeat?
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What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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bpdulog wrote:
Hi,

If we are not told can we assume that the numbers can repeat?

YES,
No mentioned on repetition of digit means that repetition of digits is allowed. and therefore all above mentioned solutions are incorrect as they assumed that repetition of digits is not allowed.

Here the solution should be smalled such number = 2222
Largest number = 5555

Sum of the pair of largest and smallest numbers = 2222+5555 = 7777

Total Such numbers = 4*4*4*4 = 256

Total suc pairs = 256/2 = 128

Sum of all pairs = 7777*128 = 995456
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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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GMATinsight wrote:
bpdulog wrote:
Hi,

If we are not told can we assume that the numbers can repeat?

YES,
No mentioned on repetition of digit means that repetition of digits is allowed. and therefore all above mentioned solutions are incorrect as they assumed that repetition of digits is not allowed.

Here the solution should be smalled such number = 2222
Largest number = 5555

Sum of the pair of largest and smallest numbers = 2222+5555 = 7777

Total Such numbers = 4*4*4*4 = 256

Total suc pairs = 256/2 = 128

Sum of all pairs = 7777*128 = 995456

GMATinsight : but sir your answer is not one of the given options? also what would be the solution in case repetition of digits is not allowed..
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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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GMATinsight wrote:
bpdulog wrote:
Hi,

If we are not told can we assume that the numbers can repeat?

YES,
No mentioned on repetition of digit means that repetition of digits is allowed. and therefore all above mentioned solutions are incorrect as they assumed that repetition of digits is not allowed.

Here the solution should be smalled such number = 2222
Largest number = 5555

Sum of the pair of largest and smallest numbers = 2222+5555 = 7777

Total Such numbers = 4*4*4*4 = 256

Total suc pairs = 256/2 = 128

Sum of all pairs = 7777*128 = 995456

If numbers cnt be repeated then in that case what will be the sum?

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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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Archit3110 wrote:
GMATinsight wrote:
bpdulog wrote:
Hi,

If we are not told can we assume that the numbers can repeat?

YES,
No mentioned on repetition of digit means that repetition of digits is allowed. and therefore all above mentioned solutions are incorrect as they assumed that repetition of digits is not allowed.

Here the solution should be smalled such number = 2222
Largest number = 5555

Sum of the pair of largest and smallest numbers = 2222+5555 = 7777

Total Such numbers = 4*4*4*4 = 256

Total suc pairs = 256/2 = 128

Sum of all pairs = 7777*128 = 995456

GMATinsight : but sir your answer is not one of the given options? also what would be the solution in case repetition of digits is not allowed..

That means that either question couldn't explain properly that repetition of digits is not allowed or Options are incoorect.

Probably the prior assumption is correct.
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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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No of ways of arranging 4- digits = No of possible 4- digit numbers = 4! = 24.
Each digit will appear for 6 times at each of thousandth, hundredth, tens and ones position.
The sum of the digits at ones place = 6(2 + 3 + 4 + 5) = 84.
Similarly, sum of digits at tens , hundredth, thousand place = 6(2 + 3 + 4 + 5) = 84.
Sum of all possible numbers = 1000 x 84 + 100 x 84 + 10 x 84 +84 = 93324.
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Re: What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?  [#permalink]

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fitzpratik wrote:
What is the sum of all the 4 digit numbers formed by digits 2,3,4,5?
A. 103324
B. 93324
C. 87645
D. 77678
E. 65432

Bunuel: How do we know that in this question digits can be repeated or not
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