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Combinatorics

# Permutations without repetition

A permutation is an arrangement, or listing, of objects in which the order is important.

How many different ways are there to arrange your first three classes if they are math, science, and language arts?

The arrangement of science, math, language arts is a permutation of math, science, language arts because the order of the classes is different.
You can use the Fundamental Counting Principle to find the number of permutations.

There are 6 possible arrangements, or permutations, of the 3 classes.

The number of permutations of n elements is:

Pn=n! (n factorial), where n is the number of elements of the set.

In how many ways may 3 books be placed next to each other on a shelf?

Since there are 3 books, the first place may be filled in 3 ways. There are then 2 books left, so that the second place many be filled in 2 ways. There is only 1 book left to fill the last place.

Since the arrangement of books on the shelf is important, it is a permutations problem.

p3=3!=3·2·1=6

Thus the books can be arranged in 6 ways.

In general, when we are given a problem involving permutations, where we are choosing r members from a set with n members and the order is important, the number of permutations is given by the expression

nPr=n · (n - 1) · (n - 2) · � · (n - r + 2) · (n - r + 1).

The first factor indicates we can choose the first member in n ways, the second factor indicates we can choose the second member in n - 1 ways once the first member has been chosen, and so on.

A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different permutations of these letters can be made if no letter is used more than once?

For the first letter, there are 5 possible choices. After that letter is chosen, there are 4 possible choices. Finally, there are 3 possible choices.

5P3=5 × 4 × 3 = 60

PERMUTATION FORMULA

In general nPr means that the number of permutations of n things taken r at a time. We can either use reasoning to solve these types of permutation problems or we can use the permutation formula.

The formula for permutation is:

A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different permutations of these letters can be made if no letter is used more than once?

The problem involves 5 things (A, B, C, D, E) taken 3 at a time.

There are 60 different permutations for the license plate.

Enter the number of elements of the set A and the computer will calculate you how many permutations without repetition of the elements are there?
Enter the number n =
Pn =