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The addition problem above shows four of the 24

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The addition problem above shows four of the 24 [#permalink] New post 20 Nov 2012, 23:53
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

82% (02:25) correct 18% (01:39) wrong based on 38 sessions
1,257
1,275
1,527
........
........
+7,521

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,5 and 7 exactly once in each integer. What is the sum of the 24 integers ?

(A) 26,996
(B) 44,406
(C) 60,444
(D) 66,660
(E) 99,990
[Reveal] Spoiler: OA

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Last edited by vomhorizon on 21 Nov 2012, 03:30, edited 1 time in total.
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Re: The addition problem above shows four of the 24 [#permalink] New post 21 Nov 2012, 02:10
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Numbers with 1 in the thousands postion = 6 --> 6*1000 = 6000
Numbers with 2 in the thousands postion = 6 --> 6*2000 = 12000
Numbers with 5 in the thousands postion = 6 --> 6*5000 = 30000
Numbers with 7 in the thousands postion = 6 --> 6*7000 = 42000
Numbers with 1 in the hundreds postion = 6 --> 6*100 = 600
Numbers with 2 in the hundreds postion = 6 --> 6*200 = 1200
Numbers with 5 in the hundreds postion = 6 --> 6*500 = 3000
Numbers with 7 in the hundreds postion = 6 --> 6*700 = 4200
Numbers with 1 in the tens postion = 6 --> 6*10 = 60
Numbers with 2 in the tens postion = 6 --> 6*20 = 120
Numbers with 5 in the tens postion = 6 --> 6*50 = 300
Numbers with 7 in the tens postion = 6 --> 6*70 = 420
Numbers with 1 in the ones postion = 6 --> 6*1 = 6
Numbers with 2 in the ones postion = 6 --> 6*2 = 12
Numbers with 5 in the ones postion = 6 --> 6*5 = 30
Numbers with 7 in the ones postion = 6 --> 6*7 = 42

Adding up everything = 6666 + 13332 + 33330 + 46662 = 99990

Answer should be E.

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Re: The addition problem above shows four of the 24 [#permalink] New post 21 Nov 2012, 03:31
This is the way i solved it :

Total no. of different combos = 24, total digits 4 therefore each digit must repeat 24/4 = 6 times in each row .. So the right most row would add up to 1x6 + 2x6 + 5x6 + 7x6 = 6+12+30+42 = 90 .. Each row would add up to 90, so 90 in the first means we have 9 that carries over and we get 0 , the second time its 90+9 and 9 stays and one 9 goes to the row to the left, so the last two digits of the SUM should be 90 (E) .. We could go on and solve the exact number but since only one answer choice has the last digits as 90 we needn't go any further..
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Re: The addition problem above shows four of the 24 [#permalink] New post 21 Nov 2012, 05:36
Just always add up the thousand digit of the 24 different combinations, than you can see that A B C D are always smaller than 90k.
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Re: The addition problem above shows four of the 24 [#permalink] New post 21 Nov 2012, 06:38
vomhorizon wrote:
1,257
1,275
1,527
........
........
+7,521

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,5 and 7 exactly once in each integer. What is the sum of the 24 integers ?

(A) 26,996
(B) 44,406
(C) 60,444
(D) 66,660
(E) 99,990


Another possible method....

STEP 1:
Total possible 4 digit integers 4*3*2*1 = 24
Each digit (1,2,5,7) at each place(ones,tens,hundreds,thousands) repeats 4 times.
so 24/4 = 6 (Each digit will repeat 6 times in each place)

STEP 2:
Sum of individual distinct digits (1,2,5,7) = 15

STEP 3:
sum of individual distinct digits * No of times each digit got repeated at each place = 15*6 =90

STEP 4 :
As the question is based on 4 digit integers
1111 * 90 = 99990

If it was 3 digit number calc
111 * 90 = 9990

Hope this helps you...

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Joined: 03 Sep 2012
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Re: The addition problem above shows four of the 24 [#permalink] New post 21 Nov 2012, 06:44
Quote:
Another possible method....

STEP 1:
Total possible 4 digit integers 4*3*2*1 = 24
Each digit (1,2,5,7) at each place(ones,tens,hundreds,thousands) repeats 4 times.
so 24/4 = 6 (Each digit will repeat 6 times in each place)

STEP 2:
Sum of individual distinct digits (1,2,5,7) = 15

STEP 3:
sum of individual distinct digits * No of times each digit got repeated at each place = 15*6 =90

STEP 4 :
As the question is based on 4 digit integers
1111 * 90 = 99990

If it was 3 digit number calc
111 * 90 = 9990

Hope this helps you...

--
Shan


Same as the way i did it...
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Re: The addition problem above shows four of the 24 [#permalink] New post 21 Nov 2012, 08:38
vomhorizon wrote:
Quote:
Another possible method....

STEP 1:
Total possible 4 digit integers 4*3*2*1 = 24
Each digit (1,2,5,7) at each place(ones,tens,hundreds,thousands) repeats 4 times.
so 24/4 = 6 (Each digit will repeat 6 times in each place)

STEP 2:
Sum of individual distinct digits (1,2,5,7) = 15

STEP 3:
sum of individual distinct digits * No of times each digit got repeated at each place = 15*6 =90

STEP 4 :
As the question is based on 4 digit integers
1111 * 90 = 99990

If it was 3 digit number calc
111 * 90 = 9990

Hope this helps you...

--
Shan


Same as the way i did it...


Hmmm great...
I learnt this from my prep center... :) :)
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Re: The addition problem above shows four of the 24 [#permalink] New post 26 Dec 2012, 23:42
vomhorizon wrote:
1,257
1,275
1,527
........
........
+7,521

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,5 and 7 exactly once in each integer. What is the sum of the 24 integers ?

(A) 26,996
(B) 44,406
(C) 60,444
(D) 66,660
(E) 99,990


I used this formula...

n!/n * sum of distinct digits * 111..1 = 4!/4 * 15 * 1111= 90 *1111 = 99990
For more explanation on that forumla: Sum of all Permutations of N Distince Digits

Answer: E
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Re: The addition problem above shows four of the 24   [#permalink] 26 Dec 2012, 23:42
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