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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # The addition problem above shows four of the 24 different integers tha

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Intern  B
Joined: 27 Jan 2019
Posts: 8
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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The solutions above are great. However, I believe this can be answered using simple logic and leveraging your answer choices.

The question tells us that there are 24 integers. Given there are 4 unique integers, we know there will be 6 of each (given the symmetry).

Here's where common sense comes handy: we know that 6 unique numbers will begin with the digit 4; therefore, they will have a minimum sum of 4*6000, or 24,000. Similarly, 6 unique numbers will begin with 3 and will have a minimum sum of 3000*6, or 18,000. Therefore, we have a minimum total sum of:

6*4,000 + 6*3,000 + 6*2,000 + 6*1,000 = 24,000 + 18,000 + 12,000 + 6,000 = 60,000. Logically, our answer will be greater than 60,000. Only one answer choice remains --> E.
VP  D
Joined: 07 Dec 2014
Posts: 1253
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

1234+4321=5555
5555/2=2777.5
2777.5*24=66,660
E
Senior Manager  P
Joined: 05 Feb 2018
Posts: 440
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

24 total/4 numbers = 6
The sum of each number will be the number's digit placement * 6
6*1(1000 + 100 + 10 + 1) = 6,666
6*2(1000 + 100 + 10 + 1) = 12 * 1111 = 13,332 [...remember, 11*12 = 132, the same idea here]
6*3(1000 + 100 + 10 + 1) = 18 * 1111 = 19,998
6*4(1000 + 100 + 10 + 1) = 24 * 1111 = 26,664

Start adding, Units digit = 6+2+8+4 = 20, so 0 + 2 carryover ... 0
Tens digit = 6+3+9+6 + 2 carryover = 26, so 6 + 2 carryover ... 60
We can see only E matches
Director  V
Joined: 24 Oct 2016
Posts: 671
GMAT 1: 670 Q46 V36 GMAT 2: 690 Q47 V38 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

PS78602.01

Method 1: Sum Each Column of Digits

We need positive integers having 4 digits.

S = __ __ __ __

Since no repetition allowed, we can make 4*3*2*1 = 24 such positive integers.

Now imagine writing these 24 numbers one below the other to add.

1234
1243
...
...
x 24 combinations

When we add them, noticing the symmetry we know that there will be 6 1's in units digits, 6 2's, 6 3's and 6 4's. So units digits will add up to (1+2+3+4)*6.

Similarly, tens digits will add up (1+2+3+4)*6*10
Similarly, hundreds digits will add up (1+2+3+4)*6*100
Similarly, thousands digits will add up (1+2+3+4)*6*100)

(1+2+3+4)*6 + (1+2+3+4)*6*10 + (1+2+3+4)*6*100 + (1+2+3+4)*6*1000 = (1+2+3+4)*6 * (1 + 10 + 100 + 1000) = 10 * 6 * 1111 = 66,660

Method 2: Direct Formula

Sum of all n digit numbers formed by n non-zero digits without repetition is:

(n−1)!∗(sum of the digits)∗(111... n times)

= (4-3)! * (1 + 2 + 3 + 4) * (1111)
= 6 * 10 * 1111
= 66,660

GMAT Club Legend  V
Joined: 11 Sep 2015
Posts: 4886
GMAT 1: 770 Q49 V46
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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Top Contributor
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

PS78602.01

Since we're adding 24 numbers, we know that:
Six numbers will be in the form 1 _ _ _
Six numbers will be in the form 2 _ _ _
Six numbers will be in the form 3 _ _ _
Six numbers will be in the form 4 _ _ _

Let's first see what the sum is when we say all 24 numbers are 1000, 2000, 3000 or 4000
The sum = (6)(1000) + (6)(2000) + (6)(3000) + (6)(4000)
= 6(1000 + 2000 + 3000 + 4000)
= 6(10,000)
= 60,000

Since the 24 numbers are actually greater than 1000, 2000, etc, we know that the actual sum must be greater than 60,000

Cheers,
Brent
_________________
Intern  S
Joined: 30 May 2017
Posts: 42
Location: India
Concentration: Finance, Strategy
Schools: Wharton, IESE, ISB, NUS
GPA: 4
WE: Engineering (Consulting)
Re: The addition problem above shows four of the 24 different integers tha  [#permalink]

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For every digit when repition is not allowed, in each unit or tens or hundreds or thousands place we will fixed combinations.
When we have four digits to choose in each place, we have 3*2*1combinations for each digit in each place to fix. Hope that part is clear.

So basically we have 6 combinations of each of 1,2,3,4 in each of unit or tens or hundreds or thousands place.

Now sum the combinations . ( 6*1+6*2+6*3+6*4) = 60 .

It means on adding each place I get 60 ..
Now it's simple addtion alogorithm that we do in primary classes.

Unit place sum is 60 and hence place 0 and carry 6 to tens place..
Tens place sum is also 60 and add the carried 6 making it 66. Place 6 and carry 6 to hundred place .
Hundreds place sum is 60 and add carried 6 making it 66 again. Place 6 and carry 6 to thousand place.

Now thousand place sum is also 60 and add carried 6 making 66 at the thousand place.

So we have 66660. HENCE THE ANSWER.

Posted from my mobile device
_________________
You say this is a problem , I say this must be an opportunity Re: The addition problem above shows four of the 24 different integers tha   [#permalink] 17 Jan 2020, 19:40

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