Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 08 Nov 2009
Posts: 31
Location: United States (AK)
Concentration: Marketing, Entrepreneurship
GPA: 3.4
WE: Engineering (Health Care)

4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
18 Oct 2010, 12:40
2
This post received KUDOS
2
This post was BOOKMARKED
Question Stats:
41% (02:39) correct
59% (02:10) wrong based on 219 sessions
HideShow timer Statistics
4 people are sitting in a 4 seat row watching a football game. At halftime they all get up. When they return, they each randomly sit down on one of the 4 chairs. What is the likelihood that none of the 4 end up sitting in the same chair that they sat in during the first half? A. 3/24 B. 9/24 C. 15/24 D. 18/24 E. 21/24
Official Answer and Stats are available only to registered users. Register/ Login.



Retired Moderator
Joined: 02 Sep 2010
Posts: 803
Location: London

Re: 4 people  sitting in a row seat [#permalink]
Show Tags
18 Oct 2010, 14:18
3
This post received KUDOS
1
This post was BOOKMARKED
eladshus wrote: 4 people are sitting in a 4 seat row watching a football game. At halftime they all get up. When they return, they each randomly sit down on one of the 4 chairs. What is the likelihood that none of the 4 end up sitting in the same chair that they sat in during the first half?
a. 3/24 b. 9/24 c. 15/24 d. 18/24 e. 21/24
(I am getting a different result than the OA  a result that even does not appear in the answer choices. Please collaborate) Ways all 4 sit in the correct seat = 1 Ways such that exactly 3 in correct seat = 0 (If there is one person in the wrong seat, then whose seat it is must also be in the wrong seat) Ways such that exactly 2 in wrong seat = C(4,2) = 6 (choose the two in wrong seat, only one way to get it wrong) Ways such that exactly 3 in wrong seat = C(4,1) x (3!  C(3,2)  1) = 4x2= 8 (choose the one in the correct seat x the ways 3 guys can be in 3 wrong seats  which is again all ways to seat 3 people  ways in which all correct (1)  ways in which 2 wrong (C(3,2)) ) So total ways to get all 4 wrong = 4!  1  0  6  8 = 9 So probability = 9/24 Answer : (b)This is a tough question, and the lieklihood of getting it on the GMAT is very low
_________________
Math writeups 1) Algebra101 2) Sequences 3) Set combinatorics 4) 3D geometry
My GMAT story
GMAT Club Premium Membership  big benefits and savings



Intern
Joined: 08 Nov 2009
Posts: 31
Location: United States (AK)
Concentration: Marketing, Entrepreneurship
GPA: 3.4
WE: Engineering (Health Care)

Re: 4 people  sitting in a row seat [#permalink]
Show Tags
18 Oct 2010, 14:47
Man  you played it! Thanks. I didn't think of all the possibilities.
My initial solution was:
3/4 probability for the first person to NOT sit on his chair 2/3 probability for the second person to NOT sit on his chair (from the remaining 3 chairs) 1/2 probability for the third person to NOT sit on his chair (from the remaining 2 chairs)
So, my final solution was: 3/4 * 2/3 * 1/2 = 1/4
But I missed some arrangements.
Thanks
+1 from me



Math Expert
Joined: 02 Sep 2009
Posts: 39695

Re: 4 people  sitting in a row seat [#permalink]
Show Tags
19 Oct 2010, 12:55



Intern
Joined: 17 Mar 2013
Posts: 7
Concentration: Finance, International Business

Re: 4 people  sitting in a row seat [#permalink]
Show Tags
25 Jul 2013, 04:40
eladshus wrote: Man  you played it! Thanks. I didn't think of all the possibilities.
My initial solution was:
3/4 probability for the first person to NOT sit on his chair 2/3 probability for the second person to NOT sit on his chair (from the remaining 3 chairs) 1/2 probability for the third person to NOT sit on his chair (from the remaining 2 chairs)
So, my final solution was: 3/4 * 2/3 * 1/2 = 1/4
But I missed some arrangements.
Thanks
+1 from me hi....I also tried this method but later realized that we are making the probability of the last one to choose wrong seat as 1 in this method. (3/4 * 2/3 * 1/2 *1) But that is not true. The first 3 guys can interchange their seats, all sit in wrong seats and the last one can still sit in his own place. We are missing something here.



Intern
Joined: 25 Jul 2013
Posts: 2

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
25 Jul 2013, 15:47
You can also solve it as 1  ((1/4)+(3/4*1/3)+(3/4*1/3*1/2)). This gives the exact same answer. The part that's in brackets is the probability of negative outcome (Someone sitting on their chair);



Director
Status: Verbal Forum Moderator
Joined: 17 Apr 2013
Posts: 612
Location: India
GMAT 1: 710 Q50 V36 GMAT 2: 750 Q51 V41 GMAT 3: 790 Q51 V49
GPA: 3.3

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
08 Oct 2013, 23:47
R0dman wrote: You can also solve it as 1  ((1/4)+(3/4*1/3)+(3/4*1/3*1/2)). This gives the exact same answer. The part that's in brackets is the probability of negative outcome (Someone sitting on their chair); I am getting (9+6)/24 as answer?Which is equal to 15/24 = 5/8Bunuel, Can you Please comment?
_________________
Like my post Send me a Kudos It is a Good manner. My Debrief: http://gmatclub.com/forum/howtoscore750and750imovedfrom710to189016.html



Math Expert
Joined: 02 Sep 2009
Posts: 39695

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
09 Oct 2013, 02:35



Intern
Joined: 25 Jul 2013
Posts: 2

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
09 Oct 2013, 06:44
honchos wrote: R0dman wrote: You can also solve it as 1  ((1/4)+(3/4*1/3)+(3/4*1/3*1/2)). This gives the exact same answer. The part that's in brackets is the probability of negative outcome (Someone sitting on their chair); I am getting (9+6)/24 as answer?Which is equal to 15/24 = 5/8Bunuel, Can you Please comment?Once again. it's "1 negative outcome". Probability of negative outcome is in brackets. So it goes (1/4+1/4+1/8) = (6/24+6/24+3/24) =15/24. Then you deduct it from 1. So basically you getting 1 15/24 = 9/24.



Intern
Joined: 21 Sep 2013
Posts: 30
Location: United States
Concentration: Finance, General Management
GMAT Date: 10252013
GPA: 3
WE: Operations (Mutual Funds and Brokerage)

Re: 4 people  sitting in a row seat [#permalink]
Show Tags
10 Oct 2013, 23:53
I dont know whether this is the right way to approach the problem , however i get the same ans. Originally seated A B C D now when after they get up and when they sit back again . 1st A has option to sit on 3 seats ( apart from his previous seat . thus he now sits on B's seat.) 2nd Similarly B has option to sit on 3 seats ( because A has already occupied B's previous seat, thus B sits on a's seat.) 3rd Now C has only 1 option to sit on D's seat . and similarly D also has one option to sit on C's seat.) hence total favourable outcomes 3*3*1*1=9 and total possible outcomes =4!=24 probability of the favourable outcome= 9/24. Please correct me if i am wrong. eladshus wrote: Man  you played it! Thanks. I didn't think of all the possibilities.
My initial solution was:
3/4 probability for the first person to NOT sit on his chair 2/3 probability for the second person to NOT sit on his chair (from the remaining 3 chairs) 1/2 probability for the third person to NOT sit on his chair (from the remaining 2 chairs)
So, my final solution was: 3/4 * 2/3 * 1/2 = 1/4
But I missed some arrangements.
Thanks
+1 from me



Intern
Joined: 14 Dec 2011
Posts: 18
Location: India
Concentration: Technology, Nonprofit
GMAT 1: 640 Q48 V29 GMAT 2: 660 Q45 V35
GPA: 3.5
WE: Information Technology (Computer Software)

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
28 Oct 2013, 18:17
eladshus wrote: 4 people are sitting in a 4 seat row watching a football game. At halftime they all get up. When they return, they each randomly sit down on one of the 4 chairs. What is the likelihood that none of the 4 end up sitting in the same chair that they sat in during the first half?
A. 3/24 B. 9/24 C. 15/24 D. 18/24 E. 21/24 There is a formula for total number of possible de arrangements: Number of ways in which n letters can be placed in n directed letters so that no letter goes into its own envelope is = n! [ (1/2!)  (1/3!) + (1/4!) ...... (1)^n (1/n!) ] In this case n=4. Therefore, total number of possible dearrangements such that none of them sit in their place = 4! [ (1/2!)  (1/3!) + (1/4!) ] = 9. Likelihood = Favorable outcomes/Total outcomes = 9/4! = 9/24.



Intern
Joined: 15 Aug 2013
Posts: 2

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
31 Oct 2013, 07:17
Hi guys, this is my first post. Here is my way to the solution: when the 4 people come back the first one has 3/4 possibilities to choose a seat different from the previous one, let's call him A and assume he seats were B used to seat before; then B for sure will be sitting on a different place, because A is now on his place, so B's possibility to choose a different seat is 1; C will have 2 choices, and just one suitable for sitting on a different seat, so C's possibility will be 1/2; D will have no choices, because C is now on his seat, so also D will have possibility 1 to sit on a different seat. 3/4 * 1 * 1/2 * 1 = 3/8 that is the same of 9/24 Is there any flaw in the above? Thanks a lot



Intern
Joined: 06 Sep 2013
Posts: 16

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
31 Oct 2013, 10:02
bugatti wrote: Hi guys, this is my first post. Here is my way to the solution: when the 4 people come back the first one has 3/4 possibilities to choose a seat different from the previous one, let's call him A and assume he seats were B used to seat before; then B for sure will be sitting on a different place, because A is now on his place, so B's possibility to choose a different seat is 1; C will have 2 choices, and just one suitable for sitting on a different seat, so C's possibility will be 1/2; D will have no choices, because C is now on his seat, so also D will have possibility 1 to sit on a different seat. 3/4 * 1 * 1/2 * 1 = 3/8 that is the same of 9/24 Is there any flaw in the above? Thanks a lot let's call him A and assume he seats were B used to seat before;, what if A doent sit where B sat?



Intern
Joined: 15 Aug 2013
Posts: 2

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
31 Oct 2013, 17:28
Quote: what if A doent sit where B sat? Yes you are right, that's the flaw. By changing the choice made by A at the beginning changes the probability as a whole. For instance if A sits where C used to seat, then B will have 2/3 of possibilities to sit on a different seat. It doesn't work



Manager
Joined: 03 Apr 2013
Posts: 179

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
17 Nov 2013, 09:39
I got the correct answer, its simple, this problem is based on the Derangement Formula. Search the net and read about it, it comes to use in many problem scenarios.
_________________
Spread some love..Like = +1 Kudos



Current Student
Joined: 06 Sep 2013
Posts: 1997
Concentration: Finance

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
02 Jun 2014, 09:20
saurabhprashar wrote: eladshus wrote: 4 people are sitting in a 4 seat row watching a football game. At halftime they all get up. When they return, they each randomly sit down on one of the 4 chairs. What is the likelihood that none of the 4 end up sitting in the same chair that they sat in during the first half?
A. 3/24 B. 9/24 C. 15/24 D. 18/24 E. 21/24 There is a formula for total number of possible de arrangements: Number of ways in which n letters can be placed in n directed letters so that no letter goes into its own envelope is = n! [ (1/2!)  (1/3!) + (1/4!) ...... (1)^n (1/n!) ] In this case n=4. Therefore, total number of possible dearrangements such that none of them sit in their place = 4! [ (1/2!)  (1/3!) + (1/4!) ] = 9. Likelihood = Favorable outcomes/Total outcomes = 9/4! = 9/24. Bunuel, on dearrangement formula, should one learn this for GMAT? Cheers! J



Math Expert
Joined: 02 Sep 2009
Posts: 39695

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
02 Jun 2014, 09:35
jlgdr wrote: saurabhprashar wrote: eladshus wrote: 4 people are sitting in a 4 seat row watching a football game. At halftime they all get up. When they return, they each randomly sit down on one of the 4 chairs. What is the likelihood that none of the 4 end up sitting in the same chair that they sat in during the first half?
A. 3/24 B. 9/24 C. 15/24 D. 18/24 E. 21/24 There is a formula for total number of possible de arrangements: Number of ways in which n letters can be placed in n directed letters so that no letter goes into its own envelope is = n! [ (1/2!)  (1/3!) + (1/4!) ...... (1)^n (1/n!) ] In this case n=4. Therefore, total number of possible dearrangements such that none of them sit in their place = 4! [ (1/2!)  (1/3!) + (1/4!) ] = 9. Likelihood = Favorable outcomes/Total outcomes = 9/4! = 9/24. Bunuel, on dearrangement formula, should one learn this for GMAT? Cheers! J No. This is out of the scope of the GMAT.
_________________
New to the Math Forum? Please read this: 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



Moderator
Joined: 20 Dec 2013
Posts: 189
Location: United States (NY)
GMAT 1: 640 Q44 V34 GMAT 2: 710 Q48 V40 GMAT 3: 720 Q49 V40
GPA: 3.16
WE: Consulting (Venture Capital)

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
02 Jun 2014, 16:59
killer question.. X  Y  Z  V < Original Arrangement (below are the potential seats each person can sit in to satisfy constraint) *  X  X  X Y  *  Y  Y Z  Z  *  Z V  V  V  * starting from the first nonunderlined X, you can pick any combo of the below letters a long as they don't occupy the same column (i.e. seat). there are 3 such arrangements for each X or 9 total (divided by 4! or 24 total possibilities)
_________________
MY GMAT BLOG  ADVICE  OPINIONS  ANALYSIS



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15978

Re: 4 people are sitting in a 4 seat row watching a football [#permalink]
Show Tags
07 Nov 2015, 11:07
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: 4 people are sitting in a 4 seat row watching a football
[#permalink]
07 Nov 2015, 11:07







