Find all School-related info fast with the new School-Specific MBA Forum

It is currently 01 Aug 2014, 23:05

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

How many integers between 1 and 10^21

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
3 KUDOS received
Forum Moderator
Forum Moderator
User avatar
Status: doing good things...
Joined: 02 Jul 2009
Posts: 1233
Concentration: Entrepreneurship, Finance
GMAT 1: Q V
GMAT 2: 690 Q49 V35
GPA: 3.77
WE: Corporate Finance (Other)
Followers: 149

Kudos [?]: 491 [3] , given: 527

GMAT ToolKit User GMAT Tests User Premium Member Reviews Badge
How many integers between 1 and 10^21 [#permalink] New post 15 Oct 2010, 23:50
3
This post received
KUDOS
1
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

55% (03:08) correct 45% (02:02) wrong based on 60 sessions
How many integers between 1 and 10^21 are such that the sum of their digits is 2?

A. 190
B. 210
C. 211
D. 230
E. 231
[Reveal] Spoiler: OA

_________________

Follow me, if you find my explanations useful.

Audaces fortuna juvat!

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 77

Kudos [?]: 452 [0], given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: How many integers between 1 and 10^21 [#permalink] New post 15 Oct 2010, 23:55
Pkit wrote:
How many integers between 1 and 10^21 (10 in 21st power) are such that the sum of their digits is 2?

a.190
b.210
c.211
d.230
e.231

Bunuel, please leave a chance for other people to solve this. :)

thanks


between 1 and 10^21 means we are talking about numbers with upto 21 digits.

Case 1
Numbers of the form 2,20,200,etc
Possible numbers = 21 (1 possibility for each number of digits)

Case 2
Numbers using the digits {1,1}
Such a number will have a minimum of 2 digits and a maximum of 21 digits, with the first digit=1.
Now, consider the k-digit number, with the first digit "1", out of the rest of the k-1 digits, we have k-1 ways to choose where to put the second 1. So k-1, k-digit numbers possible.
So total number of numbers = Summation(k-1;k=2 to k=21)=1+2+...+20=20(21)/2=210



So total number of numbers = 210+21=231

Answer is (e)
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Forum Moderator
Forum Moderator
User avatar
Status: doing good things...
Joined: 02 Jul 2009
Posts: 1233
Concentration: Entrepreneurship, Finance
GMAT 1: Q V
GMAT 2: 690 Q49 V35
GPA: 3.77
WE: Corporate Finance (Other)
Followers: 149

Kudos [?]: 491 [0], given: 527

GMAT ToolKit User GMAT Tests User Premium Member Reviews Badge
Re: How many integers between 1 and 10^21 [#permalink] New post 15 Oct 2010, 23:58
Excellent!

you are Speedy Gonzales :)
_________________

Follow me, if you find my explanations useful.

Audaces fortuna juvat!

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Senior Manager
Senior Manager
avatar
Joined: 13 Aug 2010
Posts: 316
Followers: 1

Kudos [?]: 7 [0], given: 1

GMAT Tests User
Re: How many integers between 1 and 10^21 [#permalink] New post 16 Oct 2010, 02:15
so the second condition means there are 21 positions and two number, 1 and 1 so place in then. hence we can use the combination formula 21c11 and get 210.
2 KUDOS received
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 77

Kudos [?]: 452 [2] , given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: How many integers between 1 and 10^21 [#permalink] New post 16 Oct 2010, 02:25
2
This post received
KUDOS
Yea that's a good way to think about it

But formula will be C(21,2) not C(21,11). C(21,2)=210

Posted from my mobile device Image
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 177

Kudos [?]: 903 [0], given: 235

GMAT Tests User Reviews Badge
Re: How many integers between 1 and 10^21 [#permalink] New post 16 Oct 2010, 02:49
E

For the combinations of 2's and 0's we have one 2 and 20 0's

total combinations =\frac{21!}{20!}= 21 -> because 20 0's are identical.

For the combinations of 1's and 0'2 we have two 1's and 19 0's

Total combinations = \frac{21!}{19!*2!}= 210

Total = 210+21 = 231
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned :)

Jo Bole So Nihaal , Sat Shri Akaal

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Senior Manager
Senior Manager
User avatar
Joined: 20 Jan 2010
Posts: 278
Schools: HBS, Stanford, Haas, Ross, Cornell, LBS, INSEAD, Oxford, IESE/IE
Followers: 13

Kudos [?]: 132 [0], given: 117

GMAT Tests User
Re: How many integers between 1 and 10^21 [#permalink] New post 16 Oct 2010, 04:33
Wow!! whenever I see these kinda questions I got stunned but guys like bunuel,shrouded1, and gurpreet are there to help :). Nice method gurpreet :) and whenever a quant question is posted, shrouded1 is always there to explain it. Thank you! guys (I couldn't solve it but now i know how to solve it :) )
_________________

"Don't be afraid of the space between your dreams and reality. If you can dream it, you can make it so."
Target=780
http://challengemba.blogspot.com
Kudos??

Expert Post
3 KUDOS received
Magoosh GMAT Instructor
User avatar
Joined: 28 Dec 2011
Posts: 1969
Followers: 476

Kudos [?]: 1889 [3] , given: 29

How many integers between 1 and 10^21 are such that the sum [#permalink] New post 04 Dec 2012, 13:47
3
This post received
KUDOS
Expert's post
How many integers between 1 and 10^21 are such that the sum of their digits is 2?
(A) 190
(B) 210
(C) 211
(D) 230
(E) 231


For a complete explanation, see:
http://gmat.magoosh.com/questions/835
When you submit your answer, the next page will have a complete video explanation. Each one of Magoosh's 800+ practice GMAT questions has its own video explanation, for accelerated learning.
_________________

Mike McGarry
Magoosh Test Prep

Image

Image

Moderator
Moderator
User avatar
Joined: 02 Jul 2012
Posts: 1226
Location: India
Concentration: Strategy
GMAT 1: 740 Q49 V42
GPA: 3.8
WE: Engineering (Energy and Utilities)
Followers: 61

Kudos [?]: 639 [0], given: 116

GMAT Tests User Premium Member
Re: the sum of the digits is 2 [#permalink] New post 04 Dec 2012, 20:39
Nice question..... 21C2 + 21 = 210 + 21 = 231
_________________

Did you find this post helpful?... Please let me know through the Kudos button.

Thanks To The Almighty - My GMAT Debrief

GMAT Reading Comprehension: 7 Most Common Passage Types

Manager
Manager
avatar
Joined: 26 Jul 2011
Posts: 131
Location: India
WE: Marketing (Manufacturing)
Followers: 1

Kudos [?]: 28 [0], given: 15

Re: How many integers between 1 and 10^21 are such that the sum [#permalink] New post 05 Dec 2012, 01:55
Mike and Macfauz

are'nt there 22 places to choose for placing two 1s??
Moderator
Moderator
User avatar
Joined: 02 Jul 2012
Posts: 1226
Location: India
Concentration: Strategy
GMAT 1: 740 Q49 V42
GPA: 3.8
WE: Engineering (Energy and Utilities)
Followers: 61

Kudos [?]: 639 [0], given: 116

GMAT Tests User Premium Member
Re: How many integers between 1 and 10^21 are such that the sum [#permalink] New post 05 Dec 2012, 02:13
ratinarace wrote:
Mike and Macfauz

are'nt there 22 places to choose for placing two 1s??


10^{21} has 22 places. But we are asked for "between 1 & 10^21". The greatest integer less than 10^21 will have only 21 places.
_________________

Did you find this post helpful?... Please let me know through the Kudos button.

Thanks To The Almighty - My GMAT Debrief

GMAT Reading Comprehension: 7 Most Common Passage Types

Intern
Intern
avatar
Joined: 11 Sep 2012
Posts: 7
Followers: 0

Kudos [?]: 1 [0], given: 0

Re: How many integers between 1 and 10^21 are such that the sum [#permalink] New post 28 Dec 2012, 09:10
Combinations of digits that sum up to 2 are either 1+1 or 2+0:

Combinations 1+1:

Double Digits
11 Total: 1

Triple Digits
101
110 Total: 2

Quadruple Digits
1001
1010
1100 Total: 3

So now the pattern emerges that for every N digit number, there are N-1 combinations summing up to two. The number of integers should be between 1 and 10^21, exclusive which means that the largest allowed integer does only have 21 digits, not 22 which means that the amount combinations with the largest amount of places is 20 (21-1).

We can thus calculate the total amount of integers that satisfy the question by calculating the set of consecutive integers: 1+2+3...+20 which is 210. However, we did not account for the combinations of 2 and 0 that sum up to two, we have to add them first:

Combinations 2+0:

Single Digit
2 Total: 1
Double Digits
20 Total: 1
Triple Digits
200 Total: 1

This pattern is obviously easier to comprehend and will account for an additional 21 combinations.

Hence: 210+21 = 232.

Sorry for my rusty English, I am not a native speaker :).
Intern
Intern
avatar
Joined: 28 May 2013
Posts: 27
Followers: 0

Kudos [?]: 4 [0], given: 2

How many integers between 1 and 10^21 [#permalink] New post 15 Sep 2013, 03:28
How many integers between 1 and 10^21 are such that the sum of their digits is 2?

a) 190
b) 210
c) 211
d) 230
e) 231
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 18861
Followers: 3279

Kudos [?]: 22985 [0], given: 2653

Re: How many integers between 1 and 10^21 [#permalink] New post 15 Sep 2013, 03:31
Expert's post
AkshayChittoria wrote:
How many integers between 1 and 10^21 are such that the sum of their digits is 2?

a) 190
b) 210
c) 211
d) 230
e) 231


Merging similar topics. Please refer to the solutions above.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

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; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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. ,11 Mixed Questions, 12 Fresh Meat

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., 11 New DS set.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Re: How many integers between 1 and 10^21   [#permalink] 15 Sep 2013, 03:31
    Similar topics Author Replies Last post
Similar
Topics:
How many integers between 1 and 10^21 AkshayChittoria 0 15 Sep 2013, 03:28
Experts publish their posts in the topic How many integers between 1 and 10^21 are such that the sum mikemcgarry 0 28 Dec 2012, 09:10
how many of the integers between 1 and 400 Postal 1 26 Nov 2011, 06:16
HOw many integers between 324700 and 458600 have tens 1 and puma 1 15 Jun 2008, 09:25
How many integers between 1-1000 are divisible by 11 and 35? skinsvt 3 20 May 2007, 16:11
Display posts from previous: Sort by

How many integers between 1 and 10^21

  Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Privacy Policy| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.