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Intern  Joined: 27 Mar 2012
Posts: 15
What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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11 00:00

Difficulty:   5% (low)

Question Stats: 84% (01:18) correct 16% (01:25) wrong based on 234 sessions

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What is the sum of all possible 3-digit numbers that can be constructed using the digits 3, 4 and 5 if each digit can be used only once in each number ?

A. 2660
B. 2661
C. 2662
D. 2663
E. 2664

Originally posted by sugu86 on 26 Apr 2012, 00:59.
Last edited by Bunuel on 28 Aug 2019, 22:54, edited 2 times in total.
Edited the question
Math Expert V
Joined: 02 Sep 2009
Posts: 57996
Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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11
13
sugu86 wrote:
What is the sum of all possible 3-digit numbers that can be constructed using the digits 3, 4 and 5 if each digit can be used only once in each number ?

A) 2660 B) 2661 C) 2662 D) 2663 E) 2664

Thanks,

Suganth

What is the sum of all possible 3-digit numbers that can be constructed using the digits 3, 4 and 5 if each digit can be used only once in each number ?
A. 2660
B. 2661
C. 2662
D. 2663
E. 2664

Any 3-digit number can be written as: 100a+10b+c.

# of three digit numbers with digits {3, 4, 5} is 3!=6.

These 6 numbers will have 6/3=2 times 3 as hundreds digit (a), 2 times 4 as as hundreds digit, 2 times 5 as hundreds digit.

The same with tens and units digits.

100*(2*3+2*4+2*5)+10*(2*3+2*4+2*5)+(2*3+2*4+2*5)=100*24+10*24+24=24*111=2664.

Generally the sum of all the numbers which can be formed by using the n distinct digits, is given by the formula:

(n-1)!*(sum of the digits)*(111…..n times)

In our original question: n=3. sum of digits=3+4+5=12. --> (3-1)!*(12)*(111)=24*111=2664.

Hope it's clear.
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Intern  Joined: 21 Dec 2012
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Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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13
3
Another way to approach this problem is to recognize that the way the sequence increases from the min (345) is symmetrical to the way it decreases from the max (543). Therefore if you find the average of the min and max and multiply it by the number of possibilities (3! or 6) then you'll have your answer.

$$\frac{345+543}{2} = 444$$

$$444*3! = 444*6 = 2664$$
##### General Discussion
Intern  Joined: 02 Apr 2012
Posts: 3
Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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6
sugu86 wrote:
What is the sum of all possible 3-digit numbers that can be constructed using the digits 3, 4 and 5 if each digit can be used only once in each number ?

A. 2660
B. 2661
C. 2662
D. 2663
E. 2664

the unit digits of all possible 3-digit numbers are supposed to have a sum of 3 +3 +4+4+5+5=24, so the sum of numbers should have 4 as a unit digit - 2664 is the only possible option.
Manager  Joined: 31 May 2011
Posts: 200
Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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I think Bunuel got the basic way to solve this kind of question. if the number is 4 digit or the answer has 4 number with last number is 4 then U should follow Bunuel
Intern  Joined: 02 Apr 2012
Posts: 3
Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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thaihoang305 wrote:
I think Bunuel got the basic way to solve this kind of question. if the number is 4 digit or the answer has 4 number with last number is 4 then U should follow Bunuel

Many thanks to Bunuel for his very clear explanations, I am going through all problems with his explanations in forum's PS part. For this very problem I just wanted to find out the fastest way to solve as far as you need to take time into account as well.
Manager  Joined: 04 Jan 2013
Posts: 69
Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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2
Bunuel wrote:
sugu86 wrote:
What is the sum of all possible 3-digit numbers that can be constructed using the digits 3, 4 and 5 if each digit can be used only once in each number ?

A) 2660 B) 2661 C) 2662 D) 2663 E) 2664

Thanks,

Suganth

What is the sum of all possible 3-digit numbers that can be constructed using the digits 3, 4 and 5 if each digit can be used only once in each number ?
A. 2660
B. 2661
C. 2662
D. 2663
E. 2664

Any 3-digit number can be written as: 100a+10b+c.

# of three digit numbers with digits {3, 4, 5} is 3!=6.

These 6 numbers will have 6/3=2 times 3 as hundreds digit (a), 2 times 4 as as hundreds digit, 2 times 5 as hundreds digit.

The same with tens and units digits.

100*(2*3+2*4+2*5)+10*(2*3+2*4+2*5)+(2*3+2*4+2*5)=100*24+10*24+24=24*111=2664.

Generally the sum of all the numbers which can be formed by using the n distinct digits, is given by the formula:

(n-1)!*(sum of the digits)*(111…..n times)

In our original question: n=3. sum of digits=3+4+5=12. --> (3-1)!*(12)*(111)=24*111=2664.

Hope it's clear.

wow on this type of a question i was only able to come up with the computation of possible number of ways of arranging 3 digits,but the rest part gave me problems

truly speaking @bunuel i am complete lost on this part(
These 6 numbers will have 6/3=2 times 3 as hundreds digit (a), 2 times 4 as as hundreds digit, 2 times 5 as hundreds digit.

The same with tens and units digits.

100*(2*3+2*4+2*5)+10*(2*3+2*4+2*5)+(2*3+2*4+2*5)=100*24+10*24+24=24*111=2664.)...but i guess the formula would make it easier..you should add it in the topic of number theory in the gmat math book.. Rgrds Posted from my mobile device
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Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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1
I like that approach because it has precise formula, but could you please clarify this part:

Bunuel wrote:

(n-1)!*(sum of the digits)*(111…..n times)[/b]

So when to use 111 and when to use different number? You stated 111.....n times, isn't it should be 3 times in our case?
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Math Expert V
Joined: 02 Sep 2009
Posts: 57996
Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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ziko wrote:
I like that approach because it has precise formula, but could you please clarify this part:

Bunuel wrote:

(n-1)!*(sum of the digits)*(111…..n times)[/b]

So when to use 111 and when to use different number? You stated 111.....n times, isn't it should be 3 times in our case?

111... n times mean that if we have 2 digits it should be 11, if 3 digits 111, if 4 digits it should be 1,111.

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Hope it helps.
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Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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sugu86 wrote:
What is the sum of all possible 3-digit numbers that can be constructed using the digits 3, 4 and 5 if each digit can be used only once in each number ?

A. 2660
B. 2661
C. 2662
D. 2663
E. 2664

The possible 3-digit numbers are:

345, 354, 435, 453, 543, 534

Summing just the units digits of these 6 numbers, we get:

5 + 5 + 4 + 4 + 3 + 3 = 10 + 8 + 6 = 24

The only answer with a units digit of 4 is 2,664.

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Intern  S
Joined: 20 Aug 2017
Posts: 19
Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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5
1
when we need to find the sum of digits such as in this case we can either form cases because the number of digits are less.

else the concept to find the sum is

ABC.
when C is 3,4,5 so AB can be arranged in 2! ways.

the sum of digits is 3+4+5 and since it is in the units digits it takes a place value of 1.
so it becomes (3+4+5)*1*2!

similarly in 10s digit, we fix 3,4,5 and the rest of the digits can be arranged in 2! ways.

so the place value is 10 and sum becomes (3+4+5)*10*2!

similarly for hundreds place it becomes (3+4+5)*100*2!

to find the sum of all digits we add these three,
it becomes 100(3+4+5)2! + 10(3+4+5)2! + 1(3+4+5)2!
taking 2! and sum of digits as common.

2!(3+4+5)*111

we can also generalise a formula from this,
3,4,5 ----> no of digits(n) = 3

the formula becomes
(n-1)! (111....n times) ( sum of digits)

so here n = 3.
putting the values in the formula,
(3-1)! (111) (3+4+5).

This gives us the answer as 2664.

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GMAT 1: 650 Q47 V33 What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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HullDown wrote:
What is the sum of all the possible three digit numbers that can be constructed using the digits 3,4 and 5 if each digit is used only once in each number?

A) 2664
B) 2550
C) 2320
D) 3567
E) 1456

Each no. 3, 4, 5 will occur twice at any place. So in all the places it will be 2*(3+4+5) = 24.
In the units place 4 will come and carry is 2. In the tens place it will be 24+2 = 26 so 6 and carry 2. then at hundredth place it will be again 24+2 = 26 so 6 and carry 2.

So the sum will be 2664
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What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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HullDown wrote:
What is the sum of all the possible three digit numbers that can be constructed using the digits 3,4 and 5 if each digit is used only once in each number?

A) 2664
B) 2550
C) 2320
D) 3567
E) 1456

add least and greatest:
345+543=888
888/2=444 average
444*3!=2664
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Re: What is the sum of all possible 3-digit numbers that can be constructe  [#permalink]

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HullDown wrote:
What is the sum of all the possible three digit numbers that can be constructed using the digits 3,4 and 5 if each digit is used only once in each number?

A) 2664
B) 2550
C) 2320
D) 3567
E) 1456

Number of 3 digit numbers formed using 3,4,5 = 3! with each digit appearing twice at each place.

Sum of all 6 such numbers = 2*111*(3+4+5) = 222*12= 2664

IMO E

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