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# How to solve a question like this if POE doesn't work?

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How to solve a question like this if POE doesn't work? [#permalink]  30 Dec 2009, 02:29
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Question Stats:

0% (00:00) correct 0% (00:00) wrong based on 2 sessions
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

First off, I'm not even sure if this is worded coherently. Does this mean 111, 588, and 811 are included in this set?

I figured it couldn't be "A", "B", "C", or "D" because they would be too small:

A: 158 is already > 126
B: 855 + 588 + 585 is already > 126
C: 855 + 885 + 585 + 588 + 558 + 811 + 818 is already > 126
D: ...
E: POE got me here.
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Last edited by R2I4D on 30 Dec 2009, 03:19, edited 1 time in total.
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Re: How to solve this quickly? [#permalink]  30 Dec 2009, 02:47
1
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there will 9 times 1 , 9 times 5 and nine times 8 at each place in a 3 digit no. with repetition

sum will be =100*9*(1+58) + 10*9*(1+5+8) + 1*9*(1+5+8)=13986

in order to know how it is 9 times

tot no. of ways =3*3*3=27
so in tot 27 words will be formed where each digit among 1,5,8 wil be repeated equal no. of times i.e 27/3=9
hence we have 9 , 1's ; 9 5's and 9 8's at each level
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Re: How to solve this quickly? [#permalink]  30 Dec 2009, 03:36
xcusemeplz2009 wrote:
there will 9 times 1 , 9 times 5 and nine times 8 at each place in a 3 digit no. with repetition

sum will be =100*9*(1+58) + 10*9*(1+5+8) + 1*9*(1+5+8)=13986

in order to know how it is 9 times

tot no. of ways =3*3*3=27
so in tot 27 words will be formed where each digit among 1,5,8 wil be repeated equal no. of times i.e 27/3=9
hence we have 9 , 1's ; 9 5's and 9 8's at each level

Wow. I have no idea what is happening here. Would you (or someone else) mind explaining a bit more...perhaps expounding on your method a little bit?

Perhaps, the same deal, with a two digit number using 1 and 2, with digits allowed to repeat?

How can I use xcusemeplz2009's method to arrive at the same answer?
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Re: How to solve this quickly? [#permalink]  30 Dec 2009, 05:34
2
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R2I4D wrote:
Wow. I have no idea what is happening here. Would you (or someone else) mind explaining a bit more...perhaps expounding on your method a little bit?

Perhaps, the same deal, with a two digit number using 1 and 2, with digits allowed to repeat?

How can I use xcusemeplz2009's method to arrive at the same answer?

let me try again

let the no. be xyz
Place value of no will be 100x+10y+Z.............eqn 1

TOTAL NO OF WAYS IN WHICH WE CAN MAKE A NO. WITH GIVEN CONDITION IS 3*3*3=27( as digits are getting repeated)

now at unit place i.e Z if we fix 1 , then for y we have 3 options and for x we have 3 options , so total number where unit digit is one can be formed in 3*3=9 ways

similarly for other digits at unit place can be done in 9 ways

hence at unit place a digit is getting repeated 9 times

and in the same manner in tenths place repetion for all digits will be 9 each

now the task is to find the sum
sum of digits at z place is 9*1+9*5+9*8 , WHICH
IS SAME FOR Y AND X
for that we need to put all the options in eqn 1 format
i.e 100*9[1+5+8] + 10*9[1+5+8] + 9*[1+5+8]=13986

simple way is to remeber the quick formula to find out the repetition

find out tot no. ways in which the no. can be formed and divide it by the no. of digit , then mutiply this factor with the summation of all the dig and the place value

for second ex : 1,2

two dig no with repetn can be formewd in 2*2=4 ways , repetion factor=4/2 =2 ( tot no. of ways / tot no of dig)

sum=10(place value) * 2 (rep factor) *(1+2 summation of dig)+2*3=60+6=66
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Re: How to solve a question like this if POE doesn't work? [#permalink]  01 Jan 2010, 10:06
Pretty impressive formula used by xcusemeplz2009. +1 for that!

Nevertheless the question can be solved with POE as it is quite logical....

digits in scope = 1,5,8..
with this we can get highest 3 or 4 numbers - 888, 885, 881, 858.
Adding just these 4 numbers gives us = 3512
and we have 23 more numbers to be added... Hence the correct answer would be E of course!

Cheers!
JT
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Re: How to solve a question like this if POE doesn't work?   [#permalink] 01 Jan 2010, 10:06
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