kakz 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!
Answer B. Plz help to solve.
One approach: try easier examples to determine the pattern.
What if we had just 1 letter of the alphabet?
Number of ways to place book A:
1 way.
What if we had just 2 letters of the alphabet?
Number of ways to place books A and B:
A-B
B-A
2 ways.
What if we had just 3 letters of the alphabet?
Number of ways to place books A, B and C.
A-B-C
B-A-C
B-C-A
C-B-A
4 ways.
What if we had just 4 letters of the alphabet?
Number of ways to place books A, B, C and D:
A-B-C-D
B-A-C-D
B-C-A-D
B-C-D-A
C-D-B-A
C-B-A-D
C-B-D-A
D-C-B-A
8 ways.
Every time we include another letter of the alphabet, the number of ways DOUBLES.
To illustrate why:
If we have only books A, B, and C and the first book placed on the shelf is C, we have only 1 option for the next book: B.
But if we have books A, B, C,
and D and first book placed on the shelf is C, we have TWO options for the next book: B or D.
Including another letter of the alphabet DOUBLES the number of ways.
Putting it all together:
The number of ways to place book A = 1.
Each of the 25 remaining letters will DOUBLE the number of ways.
Thus, the number of ways will DOUBLE 25 times:
1*2²�= 2²�.
The correct answer is
B.
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