GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 18 Aug 2018, 05:18

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.

Close

Request Expert Reply

Confirm Cancel

How many positive seven-digit integers include the sequence 123, in th

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47981
How many positive seven-digit integers include the sequence 123, in th  [#permalink]

Show Tags

New post 26 May 2018, 07:55
1
5
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

30% (02:00) correct 70% (02:03) wrong based on 70 sessions

HideShow timer Statistics

How many positive seven-digit integers include the sequence 123, in that order? (For example, 1234567 and 9991239)

A. 45,971
B. 46,000
C. 47,979
D. 49,961
E. 50,000

_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

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?
Extra-hard Quant Tests with Brilliant Analytics

Most Helpful Community Reply
Senior Manager
Senior Manager
User avatar
G
Joined: 24 Aug 2016
Posts: 282
Location: India
Concentration: Entrepreneurship, Operations
GMAT 1: 540 Q49 V16
GMAT 2: 640 Q47 V31
GPA: 3.4
Reviews Badge CAT Tests
Re: How many positive seven-digit integers include the sequence 123, in th  [#permalink]

Show Tags

New post 26 May 2018, 08:22
5
4
Bunuel wrote:
How many positive seven-digit integers include the sequence 123, in that order? (For example, 1234567 and 9991239)

A. 45,971
B. 46,000
C. 47,979
D. 49,961
E. 50,000


Below are the cases to consider :

a.123 _ _ _ _ : each place can be occupied by integers 0,1,2,....9 i.e., 10 ways. Hence total ways=\(10^4\)
b._ 123 _ _ _ : 1st place can be filled up with 9 ways (1,2,...9)and other 3 each in 10 ways similar to case a. Hence total ways=\(9*10^3\)
c._ _123 _ _ : Hence total ways=\(9*10^3\)
d._ _ _ 123 _ : Hence total ways=\(9*10^3\)
e._ _ _ _ 123 : Hence total ways=\(9*10^3\)

Now, there are some cases of double counting in the above:
A. 123_123: 10 ways
B.123123_ : 10 ways
C._123123 : 9 ways

Hence total unique ways = \((a+b+b+d+e)-(A+B+C)\)= \(10^4\)+\(9*10^3\)+\(9*10^3\)+\(9*10^3\)+\(9*10^3\) \(-(10+10+9)=46000-29=45971\)
Hence I will go for option A.
_________________

Please let me know if I am going in wrong direction.
Thanks in appreciation.

General Discussion
Manager
Manager
avatar
S
Joined: 25 Jul 2017
Posts: 96
How many positive seven-digit integers include the sequence 123, in th  [#permalink]

Show Tags

New post 26 May 2018, 08:34
2
Lets take "123" as one entity. All Possible combination are:
1. 123_ _ _ _ i.e for remaining places can be filled by 10*10*10*10 => 10^4 ways
2. _ 123 _ _ _ => 9 * 10*10*10 => 9*10^3 ways.
3. _ _ 123 _ _ => 9 * 10*10*10 => 9*10^3 ways.
4. _ _ _ 123 _ => 9 * 10*10*10 => 9*10^3 ways.
5. _ _ _ _ 123 => 9 * 10*10*10 => 9*10^3 ways.

Adding all => 10^4+ 4*9*10^3 => 10^3 (10+36) => 46*10^3 ways i.e. 46,000
Now, there are some cases of double counting in the above:
1. 123_123: 10 ways
2. 123123_ : 10 ways
3. _123123 : 9 ways
i.e total of 29 double counting.

Hence total unique possibilities are 46000-29 = 45,971.

A is the answer.
Manager
Manager
User avatar
S
Joined: 10 May 2018
Posts: 101
Concentration: Finance, Sustainability
Re: How many positive seven-digit integers include the sequence 123, in th  [#permalink]

Show Tags

New post 22 Jul 2018, 22:10
Woah! You make it look so easy! Definite Kudos!
anuj04 wrote:
Lets take "123" as one entity. All Possible combination are:
1. 123_ _ _ _ i.e for remaining places can be filled by 10*10*10*10 => 10^4 ways
2. _ 123 _ _ _ => 9 * 10*10*10 => 9*10^3 ways.
3. _ _ 123 _ _ => 9 * 10*10*10 => 9*10^3 ways.
4. _ _ _ 123 _ => 9 * 10*10*10 => 9*10^3 ways.
5. _ _ _ _ 123 => 9 * 10*10*10 => 9*10^3 ways.

Adding all => 10^4+ 4*9*10^3 => 10^3 (10+36) => 46*10^3 ways i.e. 46,000
Now, there are some cases of double counting in the above:
1. 123_123: 10 ways
2. 123123_ : 10 ways
3. _123123 : 9 ways
i.e total of 29 double counting.

Hence total unique possibilities are 46000-29 = 45,971.

A is the answer.

_________________

Stuck in the 600-700 score bracket? I welcome you to read my four-step course of action to a modest score.
I also invite you to critique and help me find flaws in my modus operandi. Thanks!

Re: How many positive seven-digit integers include the sequence 123, in th &nbs [#permalink] 22 Jul 2018, 22:10
Display posts from previous: Sort by

How many positive seven-digit integers include the sequence 123, in th

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  

Events & Promotions

PREV
NEXT


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

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

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®.