Author 
Message 
Intern
Joined: 27 Nov 2009
Posts: 2

In how many ways can 11 books on English and 9 books on
[#permalink]
Show Tags
27 Nov 2009, 14:40
Question Stats:
0% (00:00) correct 100% (02:05) wrong based on 18 sessions
HideShow timer Statistics
In how many ways can 11 books on English and 9 books on French be placed in a row on a shelf so that two books on French may not be together?



CEO
Joined: 17 Nov 2007
Posts: 3458
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth)  Class of 2011

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
27 Nov 2009, 16:01
1) each French book has to be separated at least by one English book, so we have following order: *FE*FE*FE*FE*FE*FE*FE*FE*F* where * denotes 10 possible places for remained 118=3 English books 2) Now, let's count in how many ways we can place remained English books: \(N = C^{10}_3+P^{10}_2+C^{10}_1 = \frac{10*9*8}{3*2} + 10*9 + 10 = 120 + 90 + 10 = 220\) where \(C^{10}_3\)  all 3 books at distinct places. \(P^{10}_2\)  2 books together and remained book at distinct place (order is important). \(C^{10}_1\)  all 3 books together. maybe there is a better solution
_________________
HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android)  The OFFICIAL GMAT CLUB PREP APP, a musthave app especially if you aim at 700+  Limited GMAT/GRE Math tutoring in Chicago



CEO
Joined: 17 Nov 2007
Posts: 3458
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth)  Class of 2011

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
27 Nov 2009, 16:08
By the way, it is a good problem of 700+ level +1
_________________
HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android)  The OFFICIAL GMAT CLUB PREP APP, a musthave app especially if you aim at 700+  Limited GMAT/GRE Math tutoring in Chicago



Math Expert
Joined: 02 Sep 2009
Posts: 49430

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
27 Nov 2009, 16:19



CEO
Joined: 17 Nov 2007
Posts: 3458
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth)  Class of 2011

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
27 Nov 2009, 16:32
I knew that there is a faster way +1
_________________
HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android)  The OFFICIAL GMAT CLUB PREP APP, a musthave app especially if you aim at 700+  Limited GMAT/GRE Math tutoring in Chicago



Intern
Joined: 27 Nov 2009
Posts: 2

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
27 Nov 2009, 17:45
Bunuel, I wish I could adopt your way to look at the problem. Sometimes, it's an easy one lost in translation. Walker, though your approach is different, but it's pretty interesting. Thanks again guys, appreciate your help here.



Senior Manager
Joined: 25 Jun 2009
Posts: 256

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
27 Nov 2009, 18:07
Wow, that was a faster way. Great explanation.



Intern
Joined: 16 Nov 2009
Posts: 4

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
28 Nov 2009, 22:23
I am not very clear guys. Buenel, in your solution you have only considered the different ways that the 9 french books can be placed in 12 position. However, even the English books can be placed in their respective slots in more than one way. i.e. let the English books be 1e, 2e, 3e etc. Now different placement of these english books can yield more possible combinations.
F 1e F 2e F 3e or F 3e F 1e F 2e or F 2e F 1e F 3e
Hope I am clear in expressing my doubt



Senior Manager
Joined: 25 Jun 2009
Posts: 256

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
29 Nov 2009, 12:48
You do bring up a good point. I always have trouble imaging how to do these problems.



Intern
Joined: 28 Sep 2009
Posts: 34
Location: Toronto, Ontario, Canada

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
29 Nov 2009, 14:23
It is true that English books may have different positions but that's not what the question is asking. It's only a matter of placing 9 books in 12 slots.



Math Expert
Joined: 02 Sep 2009
Posts: 49430

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
29 Nov 2009, 14:37
oracle wrote: I am not very clear guys. Buenel, in your solution you have only considered the different ways that the 9 french books can be placed in 12 position. However, even the English books can be placed in their respective slots in more than one way. i.e. let the English books be 1e, 2e, 3e etc. Now different placement of these english books can yield more possible combinations.
F 1e F 2e F 3e or F 3e F 1e F 2e or F 2e F 1e F 3e
Hope I am clear in expressing my doubt Think I understand your point. You are saying that along with arrangements with no French books being adjacent, English and French books themselves could be arranged in different ways. But I don't think that this is the case. Though it's quite ambiguous question in a sense. Basically when GMAT wants us to consider some items as distinct it specifies this OR it's quite obvious. In original question we don't know whether these books are distinct or not: maybe all French and English books are the same, maybe not, we don't know that. If the question were: how can we arrange 11 boys and 9 girls so that no girls are together, then the answer would be 12C9*11!*9!. As it's obvious that they are all different. If the question were: how can we arrange 11 As and 9 Bs so that no Bs are together, then the answer would be 12C9. As it's obvious that As and Bs are the same. Hope it's clear.
_________________
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? Extrahard Quant Tests with Brilliant Analytics



Senior Manager
Joined: 21 Jul 2009
Posts: 345
Schools: LBS, INSEAD, IMD, ISB  Anything with just 1 yr program.

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
29 Nov 2009, 15:11
11 english books can be arranged in 11 slots (say) in 11! ways. There will be 10 empty spaces between adjacent english books and another 2 empty spaces at either ends into which the french books can be placed such that no two french books are adjacent to one another. This is done in 12C9 ways. Total number of ways should therefore be 12C9 * 11!.
_________________
I am AWESOME and it's gonna be LEGENDARY!!!



Math Expert
Joined: 02 Sep 2009
Posts: 49430

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
29 Nov 2009, 15:56
BarneyStinson wrote: 11 english books can be arranged in 11 slots (say) in 11! ways. There will be 10 empty spaces between adjacent english books and another 2 empty spaces at either ends into which the french books can be placed such that no two french books are adjacent to one another. This is done in 12C9 ways.
Total number of ways should therefore be 12C9 * 11!. Not so. Let's say we have A1, A2, B1, B2 (meaning that As and Bs are distinct). We want to arrange them so that no Bs are adjacent: *A1*A2* and we can place B1 and B2 in 3 empty slots. It can be done in 3C2 # of ways. BUT A1 and A2 can be arranged like *A1*A2* OR *A2*A1*, plus B1 and B2 can be arranged as B1B2 or B2B1. Total # of ways 3C2*2!*2!=12. Still if not convinced: B1,A1,B2,A2 B1,A2,B2,A1 B2,A1,B1,A2 B2,A2,B1,A1 A1,B1,A2,B2 A2,B1,A1,B2 A1,B2,A2,B1 A2,B2,A1,B1, B1,A1,A2,B2 B1,A2,A1,B2 B2,A1,A2,B1 B2,A2,A1,B1 BUT again this is the case when we have DISTINCT items. So, if we were told that all French book are different and all English books are different, then the answer would be 12C9*11!*9!. In our original question we are not told that French books are different and are not told that English books are different. So # of ways would be 12C9. Let's consider the easier example: # of ways to arrange two As and 2 Bs so that no Bs are adjacent: 3C2=3. *A*A* BABA ABAB BAAB Hope it helps.
_________________
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? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 16 Nov 2009
Posts: 4

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
06 Dec 2009, 03:07
Thanks Buenel. Ya, It helps, here I was confused because it does not mention whether we should consider the English Books and the French books to be distinct, or as you said, like any A's and B's, which reduces the total number of possible arrangements.



Senior Manager
Joined: 25 Jun 2009
Posts: 256

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
06 Dec 2009, 17:43
It seems possible that this question is somewhat unclear. What is the source of this question?



Manager
Joined: 22 Jul 2009
Posts: 170
Location: Manchester UK

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
17 Dec 2009, 09:05
@Bunuel u really offer great solutions...thanks once again.



Intern
Joined: 03 Sep 2010
Posts: 15

Re: Permutation Combination bookshelf
[#permalink]
Show Tags
05 Oct 2010, 11:23
@bunuel  Just WOW!
I have never had the clearer picture of combination before .



Manager
Joined: 23 Sep 2013
Posts: 138
Concentration: Strategy, Marketing
WE: Engineering (Computer Software)

Re: In how many ways can 11 books on English and 9 books on
[#permalink]
Show Tags
17 Aug 2014, 23:35
Hi Bunuel !! In the question it is mentioned that  ' 11 books on English and 9 books on French be placed in a row '. As per the question since it says 11 books on English and 9 books on , it is implicitly saying that these books of French and English are of different subjects, therfore are distinct. I doubt if we should consider these books as similar. So the answer should be: 12P9.11! Please correct me if there is a gap in my understanding.



NonHuman User
Joined: 09 Sep 2013
Posts: 8176

Re: In how many ways can 11 books on English and 9 books on
[#permalink]
Show Tags
17 Jul 2018, 10:21
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: In how many ways can 11 books on English and 9 books on &nbs
[#permalink]
17 Jul 2018, 10:21






