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

Can you please explain your solution?I didn't understand it.

Thanks,

Hi chetan2u ,

Could you please help with this one.


Thanks

here we have to place books such that no space is there between them

so first select a book and place it in the rack (since the position is determined with respect to the first book placed)
so now after placing the first book from second book we have to places for every book ie either extreme right or extreme left
hence 2^25(as there are 25 books left)

I have a question.

It seems to me that the task will be completed in 26 steps.

Step 1: Select the first book. This can be done in 26 ways
Step 2: There are only 2 books you can select that leave no gaps, so the second book can be selected 2 ways
.
.
.
Step 26: You are at the last book, there is only 1 way this one can be selected

So why isn't the answer:

26*2^24*1

or written differently

13*2^25

?

Thank you in advance

Solution:

Instead of starting with 26 volumes, let’s look at some easier examples that will allow us to develop a pattern which will be applicable for the entire set of 26 volumes. For example, let’s assume the set of encyclopedias has only 3 volumes: A, B, and C. If Bill starts with A, then he can only put them as A-B-C. If he starts with B, then he can put them as B-A-C or as B-C-A. If he starts with C, then he can only put them as C-B-A. We see that if there are 3 volumes, there are 4 ways to put the books onto the shelf. (Note: The three letter arrangements with hyphens, B-A-C for example, means the order the books are put onto the shelf. In this case, B is the first book to put onto the shelf, A is the second and C is the third. It doesn’t mean the arrangement of the books on the shelf is BAC since, after all, when all the books are put onto the shelf, the arrangement, from left to right, should be ABC.)

Now let’s say there are 4 volumes: A, B, C and D. If Bill starts with A, then he can only put them as A-B-C-D. If he starts with B, then he can put them as B-A-C-D, B-C-A-D or as B-C-D-A. If he starts with C, then he can put them as C-B-A-D, C-B-D-A, C-D-B-A. Finally, if he starts with D, then he can only put them as D-C-B-A.We see that if there are 4 volumes, there are 8 ways to put the books onto the shelf.

From this, we can see a pattern. Let n be the number of volumes. If n = 3, the total number of ways is 4 = 2^2 = 2^(3 - 1). If n = 4, the total number of ways is 8 = 2^3 = 2^(4 - 1). Therefore, if n = 26, the total number of ways is 2^(26 - 1) = 2^25.

Alternate Solution:

Notice that the number of ways Bill can shelvethe books is the same as the number of ways he can unshelve the books without leaving any gaps. To see that, suppose B1 - B2 - B3 - … - B26 is one of the allowable ways of putting the books on shelves. Then, we can simply reverse the order and remove books from the shelf in the order B26 - B25 - … - B1. Using this fact, we will count the number of ways he can unshelve the books.

Beginning with the 26 books on the shelf, he can remove either the leftmost book or the right most book on the shelf in order to avoid leaving any gaps. He has two choices for the first book.

Once the first book is removed, he can remove either the leftmost book or the right most book from the remaining 25 books. He has two choices for the second book.

Continuing the pattern, he will have 2 choices for all but the last book. When there is only one book left at the shelf, he will have only one choice. Thus, the number of ways Bill can accomplish the task is 2^25.

Answer: B

Hope I wont face question like that

Happy learning

L

There are 26 books namely A,B,C,D---Z.

Assuming we picked first Book F and kept it on shelf. After F we picked other book say P. P needs to be placed either on left side or right side of F to arrange all books alphabetically. So for every next books there are two options right or left.

Thus total no of ways = 1 (first book) x 2(either left or right of preceding book) x 2 x 2 ----2... 25 times for remaining 25 books.

Hence the correct option is B 2^25

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