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, nPr = 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. Ex) Solve for n, given nP2 = 30
Using nPr to solve for n or r
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. Perms with Restrictions
Note: Must ALWAYS fill restrictions first!!
Ex) Using the first 10 letters of the alphabet, how many
different arrangements can you make if you must
(a) start with a vowel
(b) start and end with a vowel
(c) Not start or end with a vowel

9. Ex) Using all the letters of the word BRAINS (no reps),
how many 4-letter arrangements are possible
(a) ending in "R"
(b) with consonants only
(c) with consonants and vowels alternating

10. Perms with Cases
Ex) If there are six chairs in a row, how many ways can
3 boys and 3 girls sit if they must alternate?
Ex) How many different four-digit numbers can be made
that are even using the digits 0, 1, 2, 3, 4, 5, 6 ?

11. Ex) How many different 3-digit numbers can be made
using the digits 2, 3, 4, 5, and 0 if the number must be
divisible by 5?
Ex) How many different numbers of at least three digits
can be formed from the integers 1, 2, 3, 4, 5?

13. 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?