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A library has two books of three copies each and three other books

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Hovkial
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A library has two books of three copies each and three other books [#permalink] New post   29 Sep 2019, 10:07
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A library has two books of three copies each and three other books of two copies each. What is the number of ways in which all the books can be arranged on a shelf such that copies of the same book remain together?

(A) 120

(B) 140

(C) 160

(D) 180

(E) 320
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A
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nick1816
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A library has two books of three copies each and three other books [#permalink] New post   Updated on: 30 Sep 2019, 10:51
Number of ways of arranging two books of three copies each and three other books of two copies each such that copies of the same book remain together will be same as that of arranging 5 distinct books.

Number of ways= 5!=120

Hovkial wrote:
A library has two books of three copies each and three other books of two copies each. What is the number of ways in which all the books can be arranged on a shelf such that copies of the same book remain together?

(A) 120

(B) 140

(C) 160

(D) 180

(E) 320

Originally posted by nick1816 on 29 Sep 2019, 22:28.
Last edited by nick1816 on 30 Sep 2019, 10:51, edited 1 time in total.
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A library has two books of three copies each and three other books [#permalink] New post   30 Sep 2019, 10:44
Hovkial wrote:
A library has two books of three copies each and three other books of two copies each. What is the number of ways in which all the books can be arranged on a shelf such that copies of the same book remain together?

(A) 120

(B) 140

(C) 160

(D) 180

(E) 320


Since all copies of one type of book have to remain together, simplify the problem and consider all the copies of one type of book as one unit.

Using this simplified logic, we can see that we are asked to arrange five "units".

Five different "units" (books) can be arranged in 5! ways. Think of this as a permutation or as a combination problem where order matters.

Now, the copies within each unit can be arranged among themselves in only one way. Since all the copies are identical to each other, the order of arrangement within each unit does not make a difference. This logic can be extended to each of the five units.

Therefore, the total number of ways in which all five units with different numbers of copies each can be arranged is equal to 5! x 1 x 1 x 1 x 1 x 1 = 120 ways.

ANSWER: (A)
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Re: A library has two books of three copies each and three other books [#permalink] New post   30 Sep 2019, 17:06
total types of books ; 3+2 ; 5
arrangement of 5 books can be done in 5! ways ; 120
IMO A


Hovkial wrote:
A library has two books of three copies each and three other books of two copies each. What is the number of ways in which all the books can be arranged on a shelf such that copies of the same book remain together?

(A) 120

(B) 140

(C) 160

(D) 180

(E) 320
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Re: A library has two books of three copies each and three other books [#permalink] New post   03 Oct 2019, 12:24
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Hovkial wrote:
A library has two books of three copies each and three other books of two copies each. What is the number of ways in which all the books can be arranged on a shelf such that copies of the same book remain together?

(A) 120

(B) 140

(C) 160

(D) 180

(E) 320


We can let the two books of three copies each be A and B and the three other books of two copies each be C, D and E. So one such arrangement is AAABBBCCDDEE. However, since the 3 A’s have to be together, we can consider them as just one book. Likewise, we can consider the 3B’s as one book, and so are the 2 C’s, the 2D’s and the 2 E’s. Therefore, we have now 5 distinct books and the number of ways to arrange 5 books is 5! = 120.

Answer: A
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Re: A library has two books of three copies each and three other books [#permalink] New post   10 Jul 2021, 22:49
But we can arrange copies of the book as well, no?
I though the order of copies should matter as well, since for example we have copies of book A= C1, C2 and C3
C1C2C3
and
C2C1C3
are different arangements
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Re: A library has two books of three copies each and three other books [#permalink] New post   21 Oct 2023, 20:15
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).

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Re: A library has two books of three copies each and three other books [#permalink]
21 Oct 2023, 20:15
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