Last visit was: 02 Jun 2025, 12:37 It is currently 02 Jun 2025, 12:37

Close

GRE Prep 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 GRE 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
avatar
Intern
Intern
Joined: 12 Jul 2016
Posts: 13
Own Kudos [?]: 13 [8]
Given Kudos: 0
Send PM
avatar
Intern
Intern
Joined: 12 Jul 2016
Posts: 13
Own Kudos [?]: 13 [0]
Given Kudos: 0
Send PM
User avatar
Retired Moderator
Joined: 07 Jun 2014
Posts: 4815
Own Kudos [?]: 11478 [2]
Given Kudos: 0
GRE 1: Q167 V156
WE:Business Development (Energy and Utilities)
Send PM
Verbal Expert
Joined: 18 Apr 2015
Posts: 32213
Own Kudos [?]: 38348 [0]
Given Kudos: 26319
Send PM
Re: If four trains are rerouted over tracks such that there is a [#permalink]
Expert Reply
Good explanation but accordingly to the post the answer is 9 :eh: :eh:
avatar
Manager
Manager
Joined: 23 Jan 2016
Posts: 133
Own Kudos [?]: 211 [0]
Given Kudos: 0
Send PM
Re: If four trains are rerouted over tracks such that there is a [#permalink]
2
CreationCarnage wrote:
If four trains are rerouted over tracks such that there is a train on every track and no train is on its original track, then how many ways can the trains be rerouted?

a) 4
b) 6
c) 9
d) 16
e) 24


The question is little hard as you have nested restriction. I found easier to count it physically as the number 4 is small

Let us say that the four tracks are 1 2 3 4; Now no train should be on the original track, so the track 1 should have train number 2, 3, or 4. If track 1 has train number 2, then track 2 should have either track either train 1, 3, or 4; if track 1 has train number 2 and track 2 contains train 1, then only possibility for track 3 and track 4 is train 4 and train 3

So the possible sequences are with track 1 having train number 2 are 2, 1, 4, 3; 2, 3, 4, 1; 2, 4, 1, 3; So 3 possibilities.

Using similar logic for other possibilities

So the possible sequences are with track 1 having train number 3 are 3, 1, 4, 2; 3, 4, 2, 1; 3, 4, 1, 2; So 3 possibilities.

So the possible sequences are with track 1 having train number 4 are 4, 1, 2, 3; 4, 3, 1, 2; 4, 3, 2,1 ; So 3 possibilities.

In total 9 ways
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Own Kudos [?]: 12391 [0]
Given Kudos: 136
Send PM
Re: If four trains are rerouted over tracks such that there is a [#permalink]
1
CreationCarnage wrote:
If four trains are rerouted over tracks such that there is a train on every track and no train is on its original track, then how many ways can the trains be rerouted?

a) 4
b) 6
c) 9
d) 16
e) 24


For counting questions where the answer choices are pretty small, you should definitely consider LISTING all of the possible outcomes (as rapsjade does above).

For more on this see:
User avatar
Retired Moderator
Joined: 07 Jun 2014
Posts: 4815
Own Kudos [?]: 11478 [0]
Given Kudos: 0
GRE 1: Q167 V156
WE:Business Development (Energy and Utilities)
Send PM
Re: If four trains are rerouted over tracks such that there is a [#permalink]
2
Expert Reply
rapsjade wrote:

The question is little hard as you have nested restriction. I found easier to count it physically as the number 4 is small

Let us say that the four tracks are 1 2 3 4; Now no train should be on the original track, so the track 1 should have train number 2, 3, or 4. If track 1 has train number 2, then track 2 should have either track either train 1, 3, or 4; if track 1 has train number 2 and track 2 contains train 1, then only possibility for track 3 and track 4 is train 4 and train 3

So the possible sequences are with track 1 having train number 2 are 2, 1, 4, 3; 2, 3, 4, 1; 2, 4, 1, 3; So 3 possibilities.

Using similar logic for other possibilities

So the possible sequences are with track 1 having train number 3 are 3, 1, 4, 2; 3, 4, 2, 1; 3, 4, 1, 2; So 3 possibilities.

So the possible sequences are with track 1 having train number 4 are 4, 1, 2, 3; 4, 3, 1, 2; 4, 3, 2,1 ; So 3 possibilities.

In total 9 ways




I agree with your answer. My bad. I excluded only the case where all the trains were in their respective tracks, i.e Train 1 in track 1 and Train 2 on track 2.....

However the questions states that no train should be on its original track.

Manually counting seems to be the best method applicable.

Regards,
Verbal Expert
Joined: 18 Apr 2015
Posts: 32213
Own Kudos [?]: 38348 [0]
Given Kudos: 26319
Send PM
Re: If four trains are rerouted over tracks such that there is a [#permalink]
Expert Reply
I agree with the last post.

The question is not the top-notch. Moreover, what is not explicitly stated in the question should not be inferred. OR at most to some extent
Intern
Intern
Joined: 03 Oct 2024
Posts: 6
Own Kudos [?]: 3 [1]
Given Kudos: 273
Send PM
Re: If four trains are rerouted over tracks such that there is a [#permalink]
1
Solution
This is a classic derangement problem. A derangement is a permutation in which no element appears in its original position.

The number of derangements of $n$ elements is given by:

D(n) = n! * (1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)

For n = 4, we compute:

D(4) = 4! * [1 - 1/1! + 1/2! - 1/3! + 1/4!]
D(4) = 24 * [1 - 1 + 1/2 - 1/6 + 1/24]
D(4) = 24 * [1 - 1 + 12/24 - 4/24 + 1/24]
D(4) = 24 * [1 - 1 + 9/24]
D(4) = 24 * 9/24
D(4) = 9

Answer: (c) 9
Prep Club for GRE Bot
Re: If four trains are rerouted over tracks such that there is a [#permalink]
Moderators:
GRE Instructor
111 posts
GRE Forum Moderator
37 posts
Moderator
1132 posts
GRE Instructor
234 posts

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