Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 10 Mar 2013
Posts: 223
GMAT 1: 620 Q44 V31 GMAT 2: 690 Q47 V37 GMAT 3: 610 Q47 V28 GMAT 4: 700 Q50 V34 GMAT 5: 700 Q49 V36 GMAT 6: 690 Q48 V35 GMAT 7: 750 Q49 V42 GMAT 8: 730 Q50 V39

Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
17 Jul 2015, 19:30
Bunuel, is this valid? p(2 keys correct)=Match*Match*Miss*Miss=(1/4)*(1/3)*(1/2)*1 = 1/24 P(2 keys correct)=p(2 keys correct)*Total combinations that this event can happen Total combinations that this event can happen = Number of attempted matches that will be successful = Number of ways we can arrange Ma,Ma,Mi,Mi = C(4,2) P(2 keys correct)=1/24*C(4,2)=1/24*6=1/4



SVP
Joined: 08 Jul 2010
Posts: 2356
Location: India
GMAT: INSIGHT
WE: Education (Education)

Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
18 Jul 2015, 06:49
Quote: Each of four different locks has a matching key. The keys are randomly reassigned to the locks. What is the probability that exactly two of the keys fit the locks to which they are reassigned?
A 1/8 B. 1/6 C. 1/4 D. 3/8 E. 1/2 TooLong150 wrote: Bunuel, is this valid? p(2 keys correct)=Match*Match*Miss*Miss=(1/4)*(1/3)*(1/2)*1 = 1/24 P(2 keys correct)=p(2 keys correct)*Total combinations that this event can happen Total combinations that this event can happen = Number of attempted matches that will be successful = Number of ways we can arrange Ma,Ma,Mi,Mi = C(4,2) P(2 keys correct)=1/24*C(4,2)=1/24*6=1/4 Hi TooLong150, Your explanation is ABSOLUTELY CORRECT Just another Traditional way of solving the same question\(Probability = \frac{Favourable Outcomes}{Total Outcomes}\)Total Possible Arrangement of 4 keys to 4 locks = 4! = 24 Favourable Ways of Arrangement of Keys (2 correct and 2 incorrect) = 4C2*1 = 6 4C2 represents the ways to select the lock in which the key is found to be correct 1 is the way to arrange the remaining two keys to remaining two locks such that key doesn't match the locki.e. Probability = 6/24 = 1/4 Answer: Option C
_________________
Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha email: info@GMATinsight.com I Call us : +919999687183 / 9891333772 Online OneonOne Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html
22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION



Manager
Joined: 07 Apr 2015
Posts: 175

Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
02 Aug 2015, 01:48
FIT FIT NOFIT NOFIT \((1/4) * (1/3) * (1/2) * 1 * \frac{4!}{(2!2!)} = 1/4\)



Current Student
Joined: 15 May 2015
Posts: 37

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
04 Aug 2015, 06:03
Hi Bunuel ,
If both keys and locks are different,then isnt the total number of ways is 4^2(2 keys and 4 locks). and the number of desired cases is 2^2.
so probability = 1/4
Can you please check whether the approach is right.



Board of Directors
Joined: 17 Jul 2014
Posts: 2682
Location: United States (IL)
Concentration: Finance, Economics
GPA: 3.92
WE: General Management (Transportation)

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
26 Mar 2016, 17:51
my approach... we need to select 2 keys out of 4 to be chosen correctly. we can get this in 4C2 ways, or 6 ways. now..we have 4 doors... to unlock the first one  1/4 to unlock the second one  1/3 we are left with 2 keys..we need to make sure that they do not match probability that one key doesn't match 1/2 probability that the other one doesn't match 1. now..probability that 2 are unlocked and 2 not: 1/4 * 1/3 * 1/2 * 1 = 1/24 since there are 6 possible arrangements, multiply 1/24 with 6. 1/4
C



Director
Joined: 26 Aug 2016
Posts: 662
Location: India
Concentration: Strategy, Marketing
GMAT 1: 690 Q50 V33 GMAT 2: 700 Q50 V33 GMAT 3: 730 Q51 V38
GPA: 4
WE: Consulting (Consulting)

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
01 May 2017, 10:28
mikemcgarry wrote: Dhairya275 wrote: Each of four different locks has a matching key. The keys are randomly reassigned to the locks. What is the probability that exactly two of the keys fit the locks to which they are reassigned?
1. 1/8 2. 1/6 3. 1/4 4. 3/8 5. 1/2 Help please ! Any Simple Solution ? Dear Dhairya275There are not hugely simple solutions to this. All solutions that occur to me involve using counting techniques. You might take a look at this blog post. http://magoosh.com/gmat/2013/gmatproba ... echniques/Total number of orders for 4 keys = 4! = 4*3*2*1 = 24 Of those 24 possible orders, how many have two keys in the right place and two in the wrong place. Suppose the locks are {a, b, c, d}. If the keys are in the order {A, B, C, D}, then all four are correct. To get two right & two wrong, we would need to select one pair from {A, B, C, D} and switch them. How many different pairs can we select from a set of four? 4C2 = \(\frac{4!}{(2!)(2!)}\) = \(\frac{4*3*2*1}{(2*1)(2*1)}\) = \(\frac{4*3}{2}\) = 6 So, of the 24 sets, 6 of them would have two right & two wrong. P = 6/24 = 1/4 Does this make sense? Mike Thank you . It helped me a lot . .



Intern
Joined: 20 Aug 2017
Posts: 16

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
13 Dec 2017, 12:14
Bunuel wrote: gurpreetsingh wrote: This is first question of Mgmat challenge set of Gmat Club tests. Though while solving the tests my question was wrong, but later I tried to solve it and got the correct answer. I looked at the explanation it was too long. I have done in a simple way, but I m not 100% if I m correct. OA is C \(1/4\)
Let L1,L2,L3,L4 are locks with K1,K2,K3,K4 respective keys.
Final output after merging them is L1K1, L2K2, L3K4, L4K3. Now we have to find the probability of happening the above arrangement.
What I did was, I supposed the above arrangement to be M M U U where M  matching, U unmatching
The above can be arranged in \(4!/2!2!\) = 6
Total number of arrangement = total number of ways = 4! = 24
Hence the probability = \(\frac{6}{24} = \frac{1}{4}\) No this approach is not right (you've got the correct answer because 2 keys which should be assigned incorrectly can be assigned only in 1 way {Ab; Ba}). Consider this: if it were 5 locks instead of 4 and everything else remained the same. Your approach would give MMUUU = \(\frac{5!}{2!3!}=10\) > total # of assignments 5! > \(P=\frac{10}{120}\). But correct answer would be: \(C^2_5\)  choosing which 2 keys will fit > other 3 keys can be arranged so that no other key to fit in 2 ways: {Ab; Bc; Ca} OR {Ac; Ba; Cb}. So total # of ways to assign exactly 2 keys to fit would be \(C^2_5*2\). So \(P=\frac{C^2_5*2}{5!}=\frac{20}{120}\). Hope it's clear. Hey Bunuel, can you please tell me if theres any article that explains this C technique in the forum? thank you!



Intern
Joined: 01 Dec 2017
Posts: 14
Location: Italy
GPA: 4

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
13 Dec 2017, 13:02
GMatAspirerCA wrote: I was just going over probability questions. can some one explain me what's wrong in my approach here.
Probability of choosing one right key out of 4 is 1/4.
Probablity of choosing another right key is 1/4.
since the question is asking for 2 right keys , probability is multiplication of both = 1/4 * 1/4 = 1/16.
I went through explanations here. but this is how I solved when i looked at problem. Can some one correct me why is this approach not taken?
Thanks You have to remember that the four events ar NOT independent, therefore you can't do 1/4/*1/4....Once you got the first key right probability of the second to be right becomes 1/3



Math Expert
Joined: 02 Sep 2009
Posts: 49429

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
13 Dec 2017, 20:15
omavsp wrote: Bunuel wrote: gurpreetsingh wrote: This is first question of Mgmat challenge set of Gmat Club tests. Though while solving the tests my question was wrong, but later I tried to solve it and got the correct answer. I looked at the explanation it was too long. I have done in a simple way, but I m not 100% if I m correct. OA is C \(1/4\)
Let L1,L2,L3,L4 are locks with K1,K2,K3,K4 respective keys.
Final output after merging them is L1K1, L2K2, L3K4, L4K3. Now we have to find the probability of happening the above arrangement.
What I did was, I supposed the above arrangement to be M M U U where M  matching, U unmatching
The above can be arranged in \(4!/2!2!\) = 6
Total number of arrangement = total number of ways = 4! = 24
Hence the probability = \(\frac{6}{24} = \frac{1}{4}\) No this approach is not right (you've got the correct answer because 2 keys which should be assigned incorrectly can be assigned only in 1 way {Ab; Ba}). Consider this: if it were 5 locks instead of 4 and everything else remained the same. Your approach would give MMUUU = \(\frac{5!}{2!3!}=10\) > total # of assignments 5! > \(P=\frac{10}{120}\). But correct answer would be: \(C^2_5\)  choosing which 2 keys will fit > other 3 keys can be arranged so that no other key to fit in 2 ways: {Ab; Bc; Ca} OR {Ac; Ba; Cb}. So total # of ways to assign exactly 2 keys to fit would be \(C^2_5*2\). So \(P=\frac{C^2_5*2}{5!}=\frac{20}{120}\). Hope it's clear. Hey Bunuel, can you please tell me if theres any article that explains this C technique in the forum? thank you! 21. Combinatorics/Counting Methods For more: ALL YOU NEED FOR QUANT ! ! !Ultimate GMAT Quantitative MegathreadHope 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: 01 Dec 2017
Posts: 14
Location: Italy
GPA: 4

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
14 Dec 2017, 06:21
Bunuel wrote: gurpreetsingh wrote: Each of four different locks has a matching key. The keys are randomly reassigned to the locks. What is the probability that exactly two of the keys fit the locks to which they are reassigned?
a) \(1/8\)
b) \(1/6\)
c) \(1/4\)
d) \(3/8\)
e) \(1/2\) Total # of ways to assign the keys to the locks is \(4!\). \(C^2_4\) to choose which 2 keys will fit. Other 2 keys can be aaranged only one way. So \(P=\frac{C^2_4}{4!}=\frac{1}{4}\). Answer: C. Hi Bunuel , How would the solution change if the question was "probability of not getting ANY keys in the right lock?" I have found the following solution by enumeration, but I can't find a more analytical approach to solve this.... First we can choose three keys to couple with lock A in the wrong way : b,c,d once we chose the first key, we can always find three ways to NOT fit any of the remaining keys; for example If we choose key "b" for the first lock we will have A B C D b a d c b d a c b c d a
Therefore we have \(3*3=9\) ways to get all the keys "wrong". I think this formula can be interpreted as 3* \(C^3_2\) , but I am not convinced as why....may be it's like "in how many ways we can choose 2 keys wrong out of three"? It is also a bit strange cause in this case we already know that one key will never fit in any case....they are clearly a part of all the possible permutation, but I would like to find a more academic way to solve it.... Could you help me out with this please , Bunuel? Many many thanks!!



Math Expert
Joined: 02 Sep 2009
Posts: 49429

Re: Each of four different locks has a matching key. The keys
[#permalink]
Show Tags
14 Dec 2017, 06:24
teone83 wrote: Bunuel wrote: gurpreetsingh wrote: Each of four different locks has a matching key. The keys are randomly reassigned to the locks. What is the probability that exactly two of the keys fit the locks to which they are reassigned?
a) \(1/8\)
b) \(1/6\)
c) \(1/4\)
d) \(3/8\)
e) \(1/2\) Total # of ways to assign the keys to the locks is \(4!\). \(C^2_4\) to choose which 2 keys will fit. Other 2 keys can be aaranged only one way. So \(P=\frac{C^2_4}{4!}=\frac{1}{4}\). Answer: C. Hi Bunuel , How would the solution change if the question was "probability of not getting ANY keys in the right lock?" I have found the following solution by enumeration, but I can't find a more analytical approach to solve this.... First we can choose three keys to couple with lock A in the wrong way : b,c,d once we chose the first key, we can always find three ways to NOT fit any of the remaining keys; for example If we choose key "b" for the first lock we will have A B C D b a d c b d a c b c d a
Therefore we have \(3*3=9\) ways to get all the keys "wrong". I think this formula can be interpreted as 3* \(C^3_2\) , but I am not convinced as why....may be it's like "in how many ways we can choose 2 keys wrong out of three"? It is also a bit strange cause in this case we already know that one key will never fit in any case....they are clearly a part of all the possible permutation, but I would like to find a more academic way to solve it.... Could you help me out with this please , Bunuel? Many many thanks!! Check here: https://gmatclub.com/forum/letterarran ... 84912.html
_________________
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




Re: Each of four different locks has a matching key. The keys &nbs
[#permalink]
14 Dec 2017, 06:24



Go to page
Previous
1 2
[ 31 posts ]



