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Intern  Joined: 25 Oct 2010
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The addition problem above shows four of the 24 different integers tha  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 70% (01:58) correct 30% (02:02) wrong based on 1353 sessions

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1,234
1,243
1,324
.....
....
+4,321

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,3,4 exact;y once in each integer. What is the sum of these 24 integers?

A. 24,000
B. 26,664
C. 40,440
D. 60,000
E. 66,660

PS78602.01

Originally posted by student26 on 02 Nov 2010, 23:34.
Last edited by Bunuel on 21 Sep 2019, 14:00, edited 2 times in total.
Renamed the topic and edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 61484
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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13
1
69
Merging similar topics.

Formulas for such kind of problems (just in case):

1. Sum of all the numbers which can be formed by using the $$n$$ digits without repetition is: $$(n-1)!*(sum \ of \ the \ digits)*(111... \ n \ times)$$.

2. Sum of all the numbers which can be formed by using the $$n$$ digits (repetition being allowed) is: $$n^{n-1}*(sum \ of \ the \ digits)*(111... \ n \ times)$$.

Similar questions:
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can-someone-help-94836.html
sum-of-all-3-digit-nos-with-88864.html
permutation-88357.html
sum-of-3-digit-s-78143.html
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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20
17
student26 wrote:
1,234
1,243
1,324
.....
....
+4,321

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,3,4 exact;y once in each integer. What is the sum of these 24 integers?

A.24,000
B.26,664
C.40,440
D.60,000
E.66,660

Using the symmetry in the numbers involved (All formed using all possible combinations of 1,2,3,4), and we know there are 24 of them. We know there will be 6 each with the units digits as 1, as 2, as 3 and as 4. And the same holds true of the tens, hundreds and thousands digit.

The sum is therefore = (1 + 10 + 100 + 1000) * (1*6 +2*6 +3*6 +4*6) = 1111 * 6 * 10 = 66660

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Posts: 1516
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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12
7
1,2,3,4 can be arranged in 4! = 24 ways

The units place of all the integers will have six 1's, six 2's, six 3's and six 4's
Likewise,
The tens place of all the integers will have six 1's, six 2's, six 3's and six 4's
The hundreds place of all the integers will have six 1's, six 2's, six 3's and six 4's
The thousands place of all the integers will have six 1's, six 2's, six 3's and six 4's

Addition always start from right(UNITS) to left(THOUSANDS);

Units place addition; 6(1+2+3+4) = 60.
Unit place of the result: 0
carried over to tens place: 6

Tens place addition; 6(1+2+3+4) = 60 + 6(Carried over from Units place) = 66
Tens place of the result: 6
carried over to hunderes place: 6

Hundreds place addition; 6(1+2+3+4) = 60 + 6(Carried over from tens place) = 66
Hundreds place of the result: 6
carried over to thousands place: 6

Thousands place addition; 6(1+2+3+4) = 60 + 6(Carried over from hundreds place) = 66
Thousands place of the result: 6
carried over to ten thousands place: 6

Ten thousands place of the result: 0+6(Carried over from thousands place) = 6

Result: 66660

Ans: "E"
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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2
Bunuel wrote:
Merging similar topics.

Formulas for such kind of problems (just in case):

1. Sum of all the numbers which can be formed by using the $$n$$ digits without repetition is: $$(n-1)!*(sum \ of \ the \ digits)*(111... \ n \ times)$$.

2. Sum of all the numbers which can be formed by using the $$n$$ digits (repetition being allowed) is: $$n^{n-1}*(sum \ of \ the \ digits)*(111... \ n \ times)$$.

Similar questions:
nice-question-and-a-good-way-to-solve-103523.html
can-someone-help-94836.html
sum-of-all-3-digit-nos-with-88864.html
permutation-88357.html
sum-of-3-digit-s-78143.html

Could you tell me the way to calculate the sum which the repetition is allowed? For example: from 1,2,3,4. how can we calculate the sum of four digit number that formed from 1,2,3,4 and repetition is allowed?
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Joined: 16 Oct 2010
Posts: 10116
Location: Pune, India
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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2
hellscream wrote:

Could you tell me the way to calculate the sum which the repetition is allowed? For example: from 1,2,3,4. how can we calculate the sum of four digit number that formed from 1,2,3,4 and repetition is allowed?

The logic is no different from 'no repetition allowed' question. The only thing different is the number of numbers you can make.

How many numbers can you make using the four digits 1, 2, 3 and 4 if repetition is allowed?
You can make 4*4*4*4 = 256 numbers (there are 4 options for each digit)

1111
1112
1121
... and so on till 4444

By symmetry, each digit will appear equally in each place i.e. in unit's place, of the 256 numbers, 64 will have 1, 64 will have 2, 64 will have 3 and 64 will have 4.
Same for 10s, 100s and 1000s place.

Sum = 1000*(64*1 + 64*2 + 64*3 + 64*4) + 100*(64*1 + 64*2 + 64*3 + 64*4) + 10*(64*1 + 64*2 + 64*3 + 64*4) + 1*(64*1 + 64*2 + 64*3 + 64*4)
= (1000 + 100 + 10 + 1)(64*1 + 64*2 + 64*3 + 64*4)
= 1111*64*10 = 711040

or use the formula given by Bunuel above:
Sum of all the numbers which can be formed by using the digits (repetition being allowed) is:$$n^{n-1}$$*Sum of digits*(111...n times)
=$$4^3*(1+2+3+4)*(1111) = 711040$$ (Same calculation as above)
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Karishma
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Intern  Joined: 21 May 2013
Posts: 1
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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9
This can be solved much easier by realizing that, since the number of four term permutations is 4!, and that summing the a sequence to its reverse gives

1234 +4321 = 5555

1243 +3421 = 5555

we may see that there are 4!/2 pairings we can make, giving us

5555(12) = 66660
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GPA: 3.54
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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5
For those who could not memorize the formular, you can guess the answer in 30 secs:
Since we have 24 numbers, we will have 6 of 1 thousand something, 6 of 2 thousand something, 6 of 3 thousand something, and 6 of 4 thousand something
So,
6x1(thousand something) = 6 (thousand something)
6x2(thousand something) = 12 (thousand something)
6x3(thousand something) = 18 (thousand something)
6x4(thousand something) = 24 (thousand something)
Add them all 6+12 +18 + 24 = 60 (thousand something)
----> E
Intern  Joined: 24 Apr 2016
Posts: 6
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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each term is repeating 6 times..(total 24/4=4)
now at unit...each term will repeat 6 times... (1x6+ 2x6 + 3x6 + 4x6 = 60) , so unit digit is "0" and 6 remaining.

repeating same.....total of 24 ten digit will be 60 + 6 from total of unit ,so ten digit will be 6.

only E has 60 as last two digits.
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Concentration: Operations, Marketing
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GPA: 3.4
WE: Engineering (Manufacturing)
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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3
student26 wrote:
1,234
1,243
1,324
.....
....
+4,321

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,3,4 exact;y once in each integer. What is the sum of these 24 integers?

A. 24,000
B. 26,664
C. 40,440
D. 60,000
E. 66,660

Each digit will come at the respective place i.e units,tens, hundreds , thousands
So calculate sum of each digit for the all the places
for 4=4000+400+40+4
3=3000+300+30+3
2=2000+200+20+2
1=1000+100+10+1
Now calculate the sum of the sums of these digits =11110
Now we know that each digit is used 6 times therefore we have to multiply with 6
6*11110=66660
Hence E is our answer .
Intern  B
Joined: 16 Jul 2011
Posts: 38
Concentration: Marketing, Real Estate
GMAT 1: 550 Q37 V28
GMAT 2: 610 Q43 V31
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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VeritasPrepKarishma wrote:
hellscream wrote:

Could you tell me the way to calculate the sum which the repetition is allowed? For example: from 1,2,3,4. how can we calculate the sum of four digit number that formed from 1,2,3,4 and repetition is allowed?

The logic is no different from 'no repetition allowed' question. The only thing different is the number of numbers you can make.

How many numbers can you make using the four digits 1, 2, 3 and 4 if repetition is allowed?
You can make 4*4*4*4 = 256 numbers (there are 4 options for each digit)

1111
1112
1121
... and so on till 4444

By symmetry, each digit will appear equally in each place i.e. in unit's place, of the 256 numbers, 64 will have 1, 64 will have 2, 64 will have 3 and 64 will have 4.
Same for 10s, 100s and 1000s place.

Sum = 1000*(64*1 + 64*2 + 64*3 + 64*4) + 100*(64*1 + 64*2 + 64*3 + 64*4) + 10*(64*1 + 64*2 + 64*3 + 64*4) + 1*(64*1 + 64*2 + 64*3 + 64*4)
= (1000 + 100 + 10 + 1)(64*1 + 64*2 + 64*3 + 64*4)
= 1111*64*10 = 711040

or use the formula given by Bunuel above:
Sum of all the numbers which can be formed by using the digits (repetition being allowed) is:$$n^{n-1}$$*Sum of digits*(111...n times)
=$$4^3*(1+2+3+4)*(1111) = 711040$$ (Same calculation as above)

How about using the Average*no. of terms formula where the average is found by adding the smallest number and the largest number divided by 2. It worked for this question. My question is will it work for all such types of questions?
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Location: Pune, India
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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sam2016 wrote:
VeritasPrepKarishma wrote:
hellscream wrote:

Could you tell me the way to calculate the sum which the repetition is allowed? For example: from 1,2,3,4. how can we calculate the sum of four digit number that formed from 1,2,3,4 and repetition is allowed?

The logic is no different from 'no repetition allowed' question. The only thing different is the number of numbers you can make.

How many numbers can you make using the four digits 1, 2, 3 and 4 if repetition is allowed?
You can make 4*4*4*4 = 256 numbers (there are 4 options for each digit)

1111
1112
1121
... and so on till 4444

By symmetry, each digit will appear equally in each place i.e. in unit's place, of the 256 numbers, 64 will have 1, 64 will have 2, 64 will have 3 and 64 will have 4.
Same for 10s, 100s and 1000s place.

Sum = 1000*(64*1 + 64*2 + 64*3 + 64*4) + 100*(64*1 + 64*2 + 64*3 + 64*4) + 10*(64*1 + 64*2 + 64*3 + 64*4) + 1*(64*1 + 64*2 + 64*3 + 64*4)
= (1000 + 100 + 10 + 1)(64*1 + 64*2 + 64*3 + 64*4)
= 1111*64*10 = 711040

or use the formula given by Bunuel above:
Sum of all the numbers which can be formed by using the digits (repetition being allowed) is:$$n^{n-1}$$*Sum of digits*(111...n times)
=$$4^3*(1+2+3+4)*(1111) = 711040$$ (Same calculation as above)

How about using the Average*no. of terms formula where the average is found by adding the smallest number and the largest number divided by 2. It worked for this question. My question is will it work for all such types of questions?

Sum of terms = Average * Number of terms
holds for all sets of numbers

Average = (First term + Last term)/2
holds for an Arithmetic Progression only
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Karishma
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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VeritasPrepKarishma wrote:
sam2016 wrote:

How about using the Average*no. of terms formula where the average is found by adding the smallest number and the largest number divided by 2. It worked for this question. My question is will it work for all such types of questions?

Sum of terms = Average * Number of terms
holds for all sets of numbers

Average = (First term + Last term)/2
holds for an Arithmetic Progression only

Yes, you are right. But what was the arithmetic progression in this question?
And do such types of questions always have arithmetic progression so that I can use the above-mentioned formula for such questions?
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Joined: 16 Oct 2010
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Location: Pune, India
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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2
sam2016 wrote:
VeritasPrepKarishma wrote:
sam2016 wrote:

How about using the Average*no. of terms formula where the average is found by adding the smallest number and the largest number divided by 2. It worked for this question. My question is will it work for all such types of questions?

Sum of terms = Average * Number of terms
holds for all sets of numbers

Average = (First term + Last term)/2
holds for an Arithmetic Progression only

Yes, you are right. But what was the arithmetic progression in this question?
And do such types of questions always have arithmetic progression so that I can use the above-mentioned formula for such questions?

There isn't and hence, I haven't used this formula. Note that Avg concept will not work when the digits are say 1, 2, 4, 6.
It works in this case because of the symmetry of the digits 1, 2, 3 and 4.
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Karishma
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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4
We have 6 numbers with 1XXX, 6 with 2XXX 6 with 3XXX and 6 with 4XXX.

6*1000+6*2000+6*3000+6*4000= 60000. Hence, the solution must be higher than 60000. Only E fits.
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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What's the level of this question? +600?
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Joined: 02 Sep 2009
Posts: 61484
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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carlop wrote:
What's the level of this question? +600?

You can check difficulty level of a question in the tags above original post: The difficulty level of this question is 600-700.
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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student26 wrote:
1,234
1,243
1,324
.....
....
+4,321

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,3,4 exact;y once in each integer. What is the sum of these 24 integers?

A. 24,000
B. 26,664
C. 40,440
D. 60,000
E. 66,660

Given the information-

when we fix the thousands place say 1
sum of unit digits = 2*4+2*3+2*2 (every digit is repeated twice)
therefore
when T=1
sum = 2(2+3+4)=18
similarly
when T=2
sum = 2(1+3+4)=16
when T =3
sum =2(1+2+4)=14
when T=4
sum =2(1+2+3)=12

sum of all the units = 60
sum of all tens =60*10
sum of all hundreds =60*100
sum of all thousands=60*1000

sum of all the digits = 66660
E
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Posts: 107
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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Bunuel wrote:
Merging similar topics.

Formulas for such kind of problems (just in case):

1. Sum of all the numbers which can be formed by using the $$n$$ digits without repetition is: $$(n-1)!*(sum \ of \ the \ digits)*(111... \ n \ times)$$.

2. Sum of all the numbers which can be formed by using the $$n$$ digits (repetition being allowed) is: $$n^{n-1}*(sum \ of \ the \ digits)*(111... \ n \ times)$$.

Similar questions:
http://gmatclub.com/forum/nice-question ... 03523.html
http://gmatclub.com/forum/can-someone-help-94836.html
http://gmatclub.com/forum/sum-of-all-3- ... 88864.html
http://gmatclub.com/forum/permutation-88357.html
http://gmatclub.com/forum/sum-of-3-digit-s-78143.html

Using this formula the answers come out 71104 no choice in answer choices.
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Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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A slightly different approach:

We are looking to sum a four digit integer where each digit is not equal to zero. Therefore, taking each digit's place holder, we have a template of (1000 + 100 + 10 + 1) or (1,111).

We know that this 1,111 is repeated 24 times; therefore (1111 * 24) = 26,664.

We are unsure which digit of (1,2,3,4) goes where for any specific number within the set of 24 possibilities, but we know that the average number of (1,2,3,4) is 2.5

26,664 * 2.5 = 66,660 Re: The addition problem above shows four of the 24 different integers tha   [#permalink] 01 Apr 2019, 20:06

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