In how many ways can 11 books be arranged on a shelf?
We’re being asked to find the number of different orders or arrangements that we can place these 11 books in on the shelf. Let’s picture there’s 11 spaces on our bookshelf. And we’ll think about the number of choices that we have for each space.
The first space on our bookshelf can be filled by any of the 11 books. However, for the second space, we’ve already placed one of the books on the shelf, which means there’re only 10 choices for the book that goes in the second space. For the third book, we’ve already placed two of our 11 books on the shelf, which means there’re now nine. That’s 11 minus two books that we can choose for the third space.
By the same reasoning, there’re eight choices for the book that we put in the fourth position. The number of choices we have for the book we put in each space keeps decreasing, until we get to the final space on our shelf, by which time we have only one book left. So we have no choice at all.
To work out the number of ways our 11 books can be arranged then, we multiply together the number of choices we have for each space on the bookshelf because any of these choices can be combined with any of the others. So we have 11 times 10 times nine times eight, all the way down to one.
There is a way of abbreviating this. 11 times 10 times nine times eight times seven times six times five times four times three times two times one can be written as 11 factorial. That’s 11 followed by an exclamation mark. You may also sometimes see it written using this alternative notation. The factorial of an integer just means the product of all of the integers from one to that integer. So 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two, all the way down to one.
There’s usually a button on your calculator for evaluating factorials. And it will be either an exclamation mark or the other piece of notation, as written above. Evaluating 11 factorial on a calculator gives 39916800. If you don’t have a factorial button on your calculator, you could just type the product in longhand.
So we found that the number of ways of arranging 11 books on a shelf is 39916800.