How many ways can the librarian arrange books on the shelf?

Answer 1

It depends upon how much the librarian wants to find his books. Normally the arrangement of the books would be limited by the Dewey decimal system which classifies books into subject matter and then the shelves would be filled according to the number assigned by the system, with books of equal number being arranged by alphabetical, either by subject or author.

However, assuming a hypothetical librarian who is annoyed as people keep removing bits from his books, the books are incomplete (just like this question is incomplete as there is no indication of how many books he has) and so the librarian wants to make it difficult for the vandals to remove more bits from the same books, or is just plain bored as nobody comes into his library any more as they get all their books from Amazon, let's investigate what he can do:

If he has 1 book, he can only put in on the shelf 1 way.

But if he has 2 books, he has a choice of 2 books to put first, and for each of those choices he has one book left that he can put second, making 2 × 1 = 2 ways.

Not very exciting yet, but now think about if he has 3 books:

He can put any of the 3 first, which leaves 2 choices for the second book for each of the first book, with a final choice for the final book; in this case he has 3 × 2 × 1 = 6 ways.

By similar reasoning we can see for 4 books he can put them 4 × 3 × 2 × 1 = 24 ways;

and for 5 books it is 5 × 4 × 3 × 2 × 1 = 120 ways.

Wow! the librarian must be getting excited as the number of ways he can put his books on his shelves increases dramatically each time he adds a book.

But he has loads (and I mean loads, like hundreds) of books in his library. So how can he get this into a nice little formula? This is now going to get a bit tricky as we use some algebra, but stick with it, it'l be easy algebra:

Let's assume he has n books - n just means any number, like 1, 5, 30, 750, etc. The exact value doesn't matter for the moment.

When he places his books on the shelves, he can choose any of the n books for the first book; this leaves (n - 1) left to be placed on the shelf - this (n - 1) means that whatever value n had, subtract 1 from it, for example if he had 5 books, n = 5 and n - 1 = 5 - 1 = 4, that is after palcing the first of his 5 books he now has 4 left to place.

He can now choose any of the n-1 books left as the next book to place, which leaves (n-1)-1 = n-2 books left to place, and n × (n-1) ways so far.

Repeating this until the last 2 books, which we know he can place in 2 × 1 ways, means he can place the n books in n × (n-1) × (n-2) × ... × 2 × 1 ways.

This value, n × (n-1)× (n-2) × ... × 2 × 1, is know as "n factorial", written as n! (that is n followed by an exclamation mark).

Thus, if he has n books the librarian can arrange them in n! (n factorial) ways.

If he has just 20 books this is 20! = 2,432,902,008,176,640,000 different ways - more than enough to keep the librarian busy until his retirement.