|
Author |
Message |
|
TAGS:
|
|
|
Intern
Joined: 25 Nov 2009
Posts: 47
Location: India
Followers: 0
Kudos [?]:
5
[0], given: 6
|
Find the sum of all 3-digit nos that can be formed by 1, 2 [#permalink]
 07 Jan 2010, 02:17
Question Stats:
0% (00:00) correct
 0% (00:00) wrong based on 1 sessions
Find the sum of all 3-digit nos that can be formed by 1, 2 and 3 (Ref: Kaplan. I didn't understand the explanation there. Can u pl help?)
|
|
|
|
|
|
|
|
|
Intern
Joined: 20 Dec 2009
Posts: 14
Followers: 1
Kudos [?]:
8
[0], given: 5
|
Re: Sum of all 3-digit nos with 1, 2 & 3 [#permalink]
07 Jan 2010, 02:56
As nothing has been mentioned in the question, we'll assume that repetition of numbers are allowed.
Hence the total number of 3 digit numbers that can be formed from 1,2,3 = 3*3*3
= 27.
Now out of these 27 numbers each of the digits 1,2,3 will occur at each of the hundred's, ten's and unit's position 9 times. e.g. starting from all the numbers having 1 in hundred position we have the following numbers -
111
112
113
121
122
123
131
132
133
Similar would be the sequence for numbers starting with 2 and 3 and if we count we'll find that 1,2,3 occurs at hundreds position 9 times each; at tens position 9 times each and at units position 9 times each.
Hence the sum of all these 27 numbers =
[(1+2+3) * 100 + (1+2+3) * 10 + (1+2+3) * 1 ] * 9
= 6 * 111 * 9 = 54 * 111 = 5994.
Hope this clarifies or else I will elaborate it more.
Thanks!
|
|
|
|
|
|
Intern
Joined: 25 Nov 2009
Posts: 47
Location: India
Followers: 0
Kudos [?]:
5
[0], given: 6
|
Re: Sum of all 3-digit nos with 1, 2 & 3 [#permalink]
07 Jan 2010, 04:29
Hey thnx. It helps.
But thr's another explanation:
No of such 3-digit nos = 27 (which is Ok)
1st no = 111, Last no = 333 (these r also Ok)
Hence their average = (111 + 333)/2 = 222
(couldn't understand how this formula is applied. I thought this holds true for an AP series only)
So, Sum = Number of nos X Average of the nos
= 27 X 222 = 5994
|
|
|
|
|
|
Senior Manager
Joined: 02 Aug 2009
Posts: 274
Followers: 3
Kudos [?]:
60
[0], given: 1
|
Re: Sum of all 3-digit nos with 1, 2 & 3 [#permalink]
07 Jan 2010, 04:46
hi sudip what i can of think as a way is... total nos=3*3*3=27.....
so sum will have 27 nos .... so each no 1,2,3 will be used (27/3)9 times in each digits place (hundreds,tens and ones)
...units digit=9*(1+2+3)=54, so 4..
tens digit=9*(1+2+3)=54(+5)=9, so 9.."+5" is the carried tens digit from 54 of step 1
hundreds digit=9*(1+2+3)=54(+5)=59, so the no is 5994..
|
|
|
|
|
|
Intern
Joined: 25 Nov 2009
Posts: 47
Location: India
Followers: 0
Kudos [?]:
5
[0], given: 6
|
Re: Sum of all 3-digit nos with 1, 2 & 3 (no repeat) [#permalink]
08 Jan 2010, 01:53
Thnx.
By the way, what happens to the same problem if we are not allowed to repeat any of the digits in any particular no formed from by the digits (i.e. 111 or 221 or 133 etc are not to be considered)?
|
|
|
|
|
|
Senior Manager
Joined: 02 Aug 2009
Posts: 274
Followers: 3
Kudos [?]:
60
[0], given: 1
|
Re: Sum of all 3-digit nos with 1, 2 & 3 [#permalink]
08 Jan 2010, 03:26
if the digits are not to be repeated..
total nos=3*2*1=6.. so each no 2 times..
no is =(3+2+1)*2*100+(3+2+1)*2*10+ (3+2+1)*2*1=1200+120+12=1332
|
|
|
|
|
|
GMAT Club team member
Joined: 02 Sep 2009
Posts: 11610
Followers: 1801
Kudos [?]:
9594
[0], given: 828
|
Re: Sum of all 3-digit nos with 1, 2 & 3 [#permalink]
09 Jan 2010, 16:22
sudip135 wrote: Q. Find the sum of all 3-digit nos that can be formed by 1, 2 and 3 (Ref: Kaplan. I didn't understand the explanation there. Can u pl help?) GMAT will tell you in advance whether repetition is allowed or not. Or the wording will make it obvious. Actually there is the direct formula for this kind of problems. Of course it's better to understand the concept, then to memorize the formula but in case someone is interested here it is: 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).
_________________
PLEASE READ AND FOLLOW: 11 Rules for Posting!!!
RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory
COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!
DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!
What are GMAT Club Tests?
25 extra-hard Quant Tests
Find out what's new at GMAT Club - latest features and updates
|
|
|
|
|
|
|
Re: Sum of all 3-digit nos with 1, 2 & 3
[#permalink]
09 Jan 2010, 16:22
|
|
|
|
|
|
|
|
|
|
|
close
