Hi Guys ,
I found this Question in the WinMax preparatory material. This question is one of its kind. I have utilized many sources in my preparation for Gmat ;but I have never encountered any problem like this . I would be elated if some special entity could not only Solve but also explain an approach to solve such problems .
Q
There are 4 copies of 5 different Books .In howmany ways can they be arrangedon a shelf ?
Ans : 20 !/(4!x4!x4!x4!x4!)
Side note :I would bemore than glad if line method or any alteration of it could solve this problem .
GMAT Forum
4 postsPage 1 of 1
- hassan
Post subject: (Strange Question)There are 4 copies of 5 diff
Karan ,
Good to know that that u reached the Same answer.
But you could be more helpful to us by sharing exactly the logic that u used or the technique that u are using to answer such Questions .
Infact this question is a bit of different kind . So any insight on how to approach such question would be help Ful .
Thanks !!
I am also looking forward to have the Manhattan Gmat Staff to personally help us here !!
Good to know that that u reached the Same answer.
But you could be more helpful to us by sharing exactly the logic that u used or the technique that u are using to answer such Questions .
Infact this question is a bit of different kind . So any insight on how to approach such question would be help Ful .
Thanks !!
I am also looking forward to have the Manhattan Gmat Staff to personally help us here !!
- JonathanSchneider
- ManhattanGMAT Staff
- Posts: 370
- Joined: Sun Oct 26, 2008 3:40 pm
Hey Hassan,
Thanks for your patience. We are finally catching up in our responses.
So, the way to look at this problem is this:
We've got 20 items. These can be arranged in 20! different ways. (Proof: fo the first item, we have 20 choices, for the second item we have 19 choices, etc.)
However, some of these items are the same. That is, those items can be rearranged among the same spaces on the shelf, and the overall composition will not change. Specifically, in this instance, let's say that we have five books: A, B, C, D, and E. Now, we have four copies of book A. We can write this out AAAA. Switch around those letters and you still get the same result. How many ways are there to shuffle those four letters? Well, 4!, of course (just like the 20! ways to shuffle 20 items). So, we must divide the 20! by 4!, to eliminate all of the "recounts" of the arrangements that are essentially the same.
Of course, the same is true of the other copies of the other books: for each book we have four copies, and so for each book we must divide by 4!.
The Anagram Grid can easily represent this math:
A B C D E F G H I J K L M N O P Q R S T
A A A A B B B B C C C C D D D D E E E E
We write it this way because we have four copies each of the five different books. We now group the similar symbols on the bottom into factorials, with the total # of slots on the top as its own factorial in the numerator:
20! / (4! 4! 4! 4! 4!)
Clear?
Thanks for your patience. We are finally catching up in our responses.
So, the way to look at this problem is this:
We've got 20 items. These can be arranged in 20! different ways. (Proof: fo the first item, we have 20 choices, for the second item we have 19 choices, etc.)
However, some of these items are the same. That is, those items can be rearranged among the same spaces on the shelf, and the overall composition will not change. Specifically, in this instance, let's say that we have five books: A, B, C, D, and E. Now, we have four copies of book A. We can write this out AAAA. Switch around those letters and you still get the same result. How many ways are there to shuffle those four letters? Well, 4!, of course (just like the 20! ways to shuffle 20 items). So, we must divide the 20! by 4!, to eliminate all of the "recounts" of the arrangements that are essentially the same.
Of course, the same is true of the other copies of the other books: for each book we have four copies, and so for each book we must divide by 4!.
The Anagram Grid can easily represent this math:
A B C D E F G H I J K L M N O P Q R S T
A A A A B B B B C C C C D D D D E E E E
We write it this way because we have four copies each of the five different books. We now group the similar symbols on the bottom into factorials, with the total # of slots on the top as its own factorial in the numerator:
20! / (4! 4! 4! 4! 4!)
Clear?
4 posts Page 1 of 1