Permutations involve arranging objects or letters in a definite order. The number of permutations of n objects taken r at a time is written as nPr. To calculate nPr, use the formula nPr = n!/(n-r)!. Permutations are used to calculate possibilities like license plate numbers, phone numbers, and locker combinations. When objects are identical, the number of permutations is divided by the factorial of the number of identical objects.

1. 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

2. 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

3. Ex)
(a) How many permutations can be formed from the
letters in the word JUSTICE using all 7 letters?
In general, Pr =
n n! n r
(n - r)! n N
Using the above example..

4. (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 =

5. 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.

6. Using nPr to solve for n or r
Ex) Solve for n, given nP2 = 30
Ex) Solve for r, given 5Pr = 20

7. 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!

8. 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?