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# What is the sum of all 3 digit positive integers that can be

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VP
Joined: 07 Apr 2009
Posts: 1208
Concentration: General Management, Strategy
Schools: Duke (Fuqua) - Class of 2012
What is the sum of all 3 digit positive integers that can be  [#permalink]

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29 Apr 2009, 00:06
9
39
00:00

Difficulty:

45% (medium)

Question Stats:

69% (01:15) correct 31% (01:21) wrong based on 1060 sessions

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What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986
Math Expert
Joined: 02 Sep 2009
Posts: 51218
Re: sum of 3 digit #s  [#permalink]

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02 Feb 2011, 02:37
9
27
craky wrote:
I sowe really good formula for solving this problem in some notes downloaded from this forum. I just cannot find it, so I appologize to the author.

The formula says:

Repetition allowed:
SUM of digits * (n^n-1)*(11111 ...number composed of n 1digits)

Repetition NOT allowed:
SUM of digits * (n-1)!*(11111 ...number composed of n 1digits)

Here we have 3 digits. n is 3.
Sum of digits 1+5+8=14

Repetition allowed:
14*(3^2)*111=13986

Repetition not allowed:
14*2*111=3108

It should be:

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: $$n^{n-1}*(sum \ of \ the \ digits)*(111... \ n \ times)$$.

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Joined: 26 Apr 2009
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Re: sum of 3 digit #s  [#permalink]

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29 Apr 2009, 23:24
21
11
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

Imagine, we have got all these possible numbers written down - there are in total 3^3 numbers (each digit can be either 1 or 5 or 8)

there are 3*3 options for having a number XY1
there are 3*3 options for having a number XY5
there are 3*3 options for having a number XY8

there are 3*3 options for having a number X1Z
there are 3*3 options for having a number X5Z
there are 3*3 options for having a number X8Z

there are 3*3 options for having a number 1YZ
there are 3*3 options for having a number 5YZ
there are 3*3 options for having a number 8YZ

we can sum units, tens and hundreds independently:
summing units gives (1+5+8)*3*3
summing tens gives (1+5+8)*10*3*3
summing hundreds gives (1+5+8)*100*3*3
##### General Discussion
Intern
Joined: 26 Apr 2009
Posts: 6
Re: sum of 3 digit #s  [#permalink]

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29 Apr 2009, 05:24
12
6
E

summing
units (1+5+8)*9 +
tens (1+5+8)*9*10 +
hundreds (1+5+8)*9*100 =

= 126+1,260+12,600 = 13,986
VP
Joined: 07 Apr 2009
Posts: 1208
Concentration: General Management, Strategy
Schools: Duke (Fuqua) - Class of 2012
Re: sum of 3 digit #s  [#permalink]

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29 Apr 2009, 12:26
hi aismirnov, can you elaborated on your explanation a bit. I'm a bit weak with these types of problems. for example, what did you use tens (1+5+8)*9*10, etc.

tia
Manager
Joined: 11 Aug 2008
Posts: 126
Re: sum of 3 digit #s  [#permalink]

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14 Oct 2009, 17:47
6
I don't have your specific method but by POE I still can have E.
for 8xy alone we have 888,881,885,818,855,851,858,815,811. The total of them is larger than 7200 so there is only option E left
Manager
Joined: 14 Apr 2010
Posts: 180
Re: sum of 3 digit #s  [#permalink]

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09 Aug 2010, 20:33
this is how i approached

888
588
188
158
518
118
558
These are the number that can have 8 as the units digit. here, 8(7) = 56 so units dig is 6
similarly, for 5 as the units dig - 5(7) = 35 so units dig is 5
for 1 as the units dig - 1(7) = 7 as the unit dig.
therfore, in the sum of these dig, the units dig will be 6+5+7 = 8

Please tell me where i am wrong
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Joined: 27 Jul 2010
Posts: 157
Location: Prague
Schools: University of Economics Prague
Re: sum of 3 digit #s  [#permalink]

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02 Feb 2011, 02:15
4
2
I sowe really good formula for solving this problem in some notes downloaded from this forum. I just cannot find it, so I appologize to the author.

The formula says:

Repetition allowed:
SUM of digits * (n^n-1)*(11111 ...number composed of n 1digits)

Repetition NOT allowed:
SUM of digits * (n-1)!*(11111 ...number composed of n 1digits)

Here we have 3 digits. n is 3.
Sum of digits 1+5+8=14

Repetition allowed:
14*(3^2)*111=13986

Repetition not allowed:
14*2*111=3108
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Senior Manager
Joined: 13 Aug 2012
Posts: 429
Concentration: Marketing, Finance
GPA: 3.23
What is the sum of all 3 digit positive integers that can be  [#permalink]

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20 Dec 2012, 03:15
1
iwillwin wrote:
What is the sum of all 3 digit positive numbers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126

B. 1386

C. 3108

D. 308

E. 13986

Here is a formula to know the sum of possible arrangements when a digit is not allowed to repeat:

$$(n-1)!*sumofdigits*111 = (3-1)!*(1+5+8)*111=28*111=3108$$

But we know that digits are allowed to repeat. Thus, sum is much greater than 3108.

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Joined: 02 Nov 2012
Posts: 31
Re: What is the sum of all 3 digit positive integers that can be  [#permalink]

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04 Jan 2013, 05:50
I approach this particular problem without the formulae. Can somebody please help me if this is correct --> If you know that the numbers are allowed to repeat then the possible numbers are 3*3*3 = 27 (instead of 3*2*1 when repetition is not allowed), then you know that there will be 9 ones, 9 fives, 9 eights. So for the first position you can have the 9+45+72 = 12600, then all the answer choices will fall except for E. If you calculate further you get 12600 + 01260 + 00126 = 13,986. Bunuel, Karishma or someone else can you please confirm if this is correct?
Manager
Joined: 31 May 2012
Posts: 116
Re: What is the sum of all 3 digit positive integers that can be  [#permalink]

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04 Jan 2013, 06:43
7
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

One quickest way to answer this question !!

As we are using digits 1,5, 8 and digits are allowed to repeat. Each of the unit, tenth and hundredth digit can be used by each of three digits.
So, Total possible numbers with these digits=3 X 3 X 3 =27.

First, As we have 27 three digit number, Sum will be for sure more than 2700.. Eliminate options A,B,D

Second, If you imagine numbers with the given digits 1,5,8. We have numbers like 888,885,855,858,851. Sum is for sure more than 4000. Eliminate option C.

You are left with answer E.

----------------
consider giving a +kudo if this helps
Intern
Joined: 11 Sep 2012
Posts: 7
Re: What is the sum of all 3 digit positive integers that can be  [#permalink]

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05 Jan 2013, 08:58
1
1, 5 and 8 are allowed to be used 9 times as hundreds, tenths and units digit.

So you can line up:

9x100
9x 10
9x 1
9x500
9x 50
9x 5
9x800
9x 80
9x 8

When lining these up, you should quickly realize that it's bigger than 10.000 and pick your answer without going further.
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Posts: 5887
City: Pune
Re: What is the sum of all 3 digit positive integers that can be  [#permalink]

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05 Jan 2013, 09:40
5
2
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

Answer to this question is easier to guess than to calculate. e.g. if we take 8 at hundreds place we would get at least 9 nos. So 800 * 9 = 7200 which surpasses every option but E.

Thru conventional method
(1+5+8)9 = 126
(1+5+8)9*10=1260
(1+5+8)9*100=12600

126 + 1260 + 12600 = 13896. E
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Re: sum of 3 digit #s  [#permalink]

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01 Oct 2013, 17:44
aismirnov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

Imagine, we have got all these possible numbers written down - there are in total 3^3 numbers (each digit can be either 1 or 5 or 8)

there are 3*3 options for having a number XY1
there are 3*3 options for having a number XY5
there are 3*3 options for having a number XY8

there are 3*3 options for having a number X1Z
there are 3*3 options for having a number X5Z
there are 3*3 options for having a number X8Z

there are 3*3 options for having a number 1YZ
there are 3*3 options for having a number 5YZ
there are 3*3 options for having a number 8YZ

we can sum units, tens and hundreds independently:
summing units gives (1+5+8)*3*3
summing tens gives (1+5+8)*10*3*3
summing hundreds gives (1+5+8)*100*3*3

How did you know how to do this? I mean, how did you learn? I have 4 weeks to go until my GMAT and I haven't gotten any better at these. I have no idea how to even begin approaching these problems, and none of these formulas make any sense to me.
Math Expert
Joined: 02 Sep 2009
Posts: 51218
Re: sum of 3 digit #s  [#permalink]

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02 Oct 2013, 02:24
1
4
AccipiterQ wrote:
aismirnov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

Imagine, we have got all these possible numbers written down - there are in total 3^3 numbers (each digit can be either 1 or 5 or 8)

there are 3*3 options for having a number XY1
there are 3*3 options for having a number XY5
there are 3*3 options for having a number XY8

there are 3*3 options for having a number X1Z
there are 3*3 options for having a number X5Z
there are 3*3 options for having a number X8Z

there are 3*3 options for having a number 1YZ
there are 3*3 options for having a number 5YZ
there are 3*3 options for having a number 8YZ

we can sum units, tens and hundreds independently:
summing units gives (1+5+8)*3*3
summing tens gives (1+5+8)*10*3*3
summing hundreds gives (1+5+8)*100*3*3

How did you know how to do this? I mean, how did you learn? I have 4 weeks to go until my GMAT and I haven't gotten any better at these. I have no idea how to even begin approaching these problems, and none of these formulas make any sense to me.

Direct formulas are here: what-is-the-sum-of-all-3-digit-positive-integers-that-can-be-78143.html#p862674 Please ask if anything there is unclear.

Similar questions to practice:
find-the-sum-of-all-the-four-digit-numbers-formed-using-the-103523.html
find-the-sum-of-all-the-four-digit-numbers-which-are-formed-88357.html
find-the-sum-of-all-3-digit-nos-that-can-be-formed-by-88864.html
if-the-three-unique-positive-digits-a-b-and-c-are-arranged-143836.html
what-is-the-sum-of-all-3-digit-positive-integers-that-can-be-78143.html
what-is-the-sum-of-all-4-digit-numbers-that-can-be-formed-94836.html
the-sum-of-the-digits-of-64-279-what-is-the-141460.html
there-are-24-different-four-digit-integers-than-can-be-141891.html

Hope this helps.
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Re: What is the sum of all 3 digit positive integers that can be  [#permalink]

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27 Nov 2018, 04:34
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

I used a different method to solve the problem. Hence wanted to know if this method is correct and can be applied to problems similar to this or if it was just a one-off case. My method is as follows-

One number from 1,5 and 8 can be picked in 3C1 ways. Since repetition of numbers is allowed, the remaining numbers can also be picked in 3C1 ways.

Hence the Total no. of ways of picking 1,5,8 is 3*3C1 which equals to 18.

The Smallest no. which can be formed from digits 1,5,8 is 111 while the largest in 888. This gives a range of 777.

Hence sum of all no. formed using 1,5,8 is 18*777= 13986 (Option E).

Kindly share your thoughts on this.
VP
Joined: 07 Dec 2014
Posts: 1128
What is the sum of all 3 digit positive integers that can be  [#permalink]

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28 Nov 2018, 11:57
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

sum all possibilties without repetiition:
158+185+518+581+815+851=3108
take average:
3108/6=518
multiply average by possibilities with repetition:
518*3^3=13986
E
VP
Joined: 09 Mar 2016
Posts: 1234
What is the sum of all 3 digit positive integers that can be  [#permalink]

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28 Nov 2018, 13:08
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

i wrote following numbers and added up

111+
115+
155+
555+
118+
188+
888+

= 2,140

so i eliminated A, B and D

since there were left such numbers as 881 and 551 clearly C option was out as well without further calculating

So i picked E
What is the sum of all 3 digit positive integers that can be &nbs [#permalink] 28 Nov 2018, 13:08
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