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# Bill has a set of encyclopedias with 26 volumes, one per letter of the

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Math Expert
Joined: 02 Sep 2009
Posts: 59573
Bill has a set of encyclopedias with 26 volumes, one per letter of the  [#permalink]

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27 Nov 2019, 01:58
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45% (01:10) correct 55% (01:28) wrong based on 11 sessions

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Bill has a set of encyclopedias with 26 volumes, one per letter of the alphabet. He has a special shelf built for them with 26 slots in a row, each labeled alphabetically. After moving to a new house, Bill is faced with the task of putting the books back in their proper slots. He decides to do it in such a way that, during the process, there is never a gap between any of the books that are on the shelf. That is, each book he puts back after the first one is adjacent to a book that is already on the shelf. If he can start with any of the 26 books, how many ways can Bill accomplish his task?

A. $$26!$$

B. $$2^{25}$$

C. $$24!$$

D. $$26^2$$

E. $$14!12!$$

Are You Up For the Challenge: 700 Level Questions
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Joined: 19 Oct 2018
Posts: 1151
Location: India
Bill has a set of encyclopedias with 26 volumes, one per letter of the  [#permalink]

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27 Nov 2019, 06:03
Total number of ways
= $$\frac{(n-1)!}{(n-1)!}+\frac{(n-1)!}{1!(n-2)!}+........+\frac{(n-1)!}{(n-2)!1!}+\frac{(n-1)!}{(n-1)!}$$
= $$2^{n-1}$$
=$$2^{25}$$

Bunuel wrote:
Bill has a set of encyclopedias with 26 volumes, one per letter of the alphabet. He has a special shelf built for them with 26 slots in a row, each labeled alphabetically. After moving to a new house, Bill is faced with the task of putting the books back in their proper slots. He decides to do it in such a way that, during the process, there is never a gap between any of the books that are on the shelf. That is, each book he puts back after the first one is adjacent to a book that is already on the shelf. If he can start with any of the 26 books, how many ways can Bill accomplish his task?

A. $$26!$$

B. $$2^{25}$$

C. $$24!$$

D. $$26^2$$

E. $$14!12!$$

Are You Up For the Challenge: 700 Level Questions
Bill has a set of encyclopedias with 26 volumes, one per letter of the   [#permalink] 27 Nov 2019, 06:03
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