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Permutation when all the Objects are Distinct


Theorem - The number of permutations of n distinct elements taken r (1 r n) at a time.
The following result gives us the number of ways of arranging n distinct elements taken r at a time. 
The number of permutations of n distinct elements taken r (1 r n) at a time is
  

Proof 
For the r positions, we have the following r possibilities:

1st position: Any of the n elements.
2nd position: Any of the remaining (n - 1) elements.
3rd position: Any of the remaining (n - 2) elements.
4th position: Any of the remaining (n - 3) elements........rth position: Any of the remaining (n - r + 1)  elements
Using the product rule, we multiply these numbers to obtain, nPr, the number of ways of arranging n elements taken r at a time is


Illustration
Pooja has five distinct books. In how many different ways can she arrange these books on a shelf?

Solution
For the five books, we consider the following possibilities.

1st place: Any of the five books.
2nd place: Any of the remaining four books.
3rdplace: Any of the remaining three books.
4th place: Any of the remaining two books.
5th place: The one remaining book.
Multiplying these five number together, we find the total number of orders for the books to be5.4.3.2.1 = 5! = 120We now generalise the result obtained in the above illustration to conclude that the number of permutations (orderings) of n distinct elements is n!

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 in choosing (and ordering) 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.




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