Permutations

An arrangement of all or part of a number of things in a definite order.

Order of the Arrangement Counts!!!

Permutations are used to generate:- license plate numbers- phone numbers- social insurance numbers- locker combinations

We show permutations with the symbol "nPr" which reads

the number of permutations of "n" things arranged or taken or picked "r" at a time

For example, How many 2-letter "words" can be made using the letters of the alphabet?

Two solutions possible

Ex)(a) How many permutations can be formed from the letters in the word JUSTICE using all 7 letters?

In general, nPr = n! n r (n - r)! n N

Using the above example..

(b) How many permutations can be formed from JUSTICE using only 5 letters at a time?

or 7P5 says "using 7 objects picking or arranging them 5 at a time"

Also reads "7 pick 5"

On calculator 7P5 =

Or 7P5 =

Ex)A group of 9 different books are to be selected and arranged on a shelf for display. How many arrangements are possible?

or

9P4 =

Note: Arrangement, lineup, selected suggest order or perms.

Ex) Solve for n, given nP2 = 30

Using nPr to solve for n or r

Ex) Solve for r, given 5Pr = 20

Perms with Repeating Objects

Ex) How many ways can you arrange the letters of "WOW"?

We can only tell 3 apart.The other 3 are not distinguishable.

Since the W repeats twice, we correct by dividing by 2!

Ex) How many ways can you arrange the letters of WOWW?

In general, the number of perms of n objects with r identical objects is n!

r!

Ex) Using all letters in the AARDVARK how many different arrangements are possible?