What is Permutation?

In this section we shall determine the number of ways in which we arrange (in order) n distinct elements. We call an arrangement (in order) of n elements a permutation of n distinct elements.

A permutation of n elements is an ordering of the elements such that one element is first; one is second, and so on.
a1, a2, ..., an

For instance, 3, 2, 5, 1, 4 is a permutation of 1, 2, 3, 4, 5. As said earlier, we are interested in the number of different ways in which we can arrange n elements. To demonstrate the argument we begin with the following illustration.

Now, let us return to our earlier illustration.

Suppose Pooja has 5 distinct books on mathematics but she has room only for 3 books on the shelf.
In this case, we have the following possibilities.

1st place: Any of the five books.
2nd place: Any of the remaining four books.
3rd place: Any of the remaining three books.

Using the product rule, we multiply these three numbers together to obtain that there are 5. 4. 3. = 60 ways to arrange the books.

As seen in the above illustration, sometimes, we are interested in ordering a subset of a collection of elements rather than the entire collection. For instance, in the above illustration, Pooja was interested to choose (and order) 3 books out of 5 distinct books. In general, we shall be interested in choosing and arranging r elements out of a collection of n elements. We call such an arrangement a permutation of n elements taken r at a time.