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Pre Calculus
Permutations and Combinations
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The Counting Principle Determining the number of ways events can occur.
How many different outfits are possible if a person owns 3 pants (black, brown and blue) and 4 shirts
(white, yellow, pink and green).
If one event can occur in m ways. A second event occurs in n ways. The pair can occur in: m · n ways.
For multiple events occurring together, with m1 , m2 , m3 , m4 , m5 ,... distinct ways of occurring
respectively, the total number of distinct ways is: m1  m2  m3  m4  m5 ...
Example:
How many different outfits are possible with 4 pants, 5 shirts and 2 ties?
Permutation: An ordered arrangement of distinct objects.
How many different ways can you arrange 5 posters in a line?
For n distinct objects the number of permutations is n!
Permutations with and without repeats
Example:
In how many ways can the letters of the word LOVE be arranged?
How many 4 letter codes (passwords) can be made with letters of the word LOVE?
For m positions, n repeatable objects can be arranged in nm ways
Example: How many 7 digit telephone numbers are there, if the number must start with 7 or 5?
Evaluate the permutations for the following:
1. The number of 4 digit numbers divisible by 5.
2. The number of ways to respond to 5 true / false questions.
3. The number of ways to rearrange the letters of BREAD, if it must start with a vowel.
4. The number of codes with the letters of BREAD, starting and ending with vowels.
5. The number of ways to completely fasten a 4 button coat.
6. The number of 4 digit numbers where repeats are not allowed.
Permutations of Indistinguishable Objects
Example: How many distinct permutations of the word BANANA are there?
Distinguishable
Permutations of n objects, of which k1 , k 2 , k 3 are identical 
n!
k1 ! k 2 ! k 3 !
How many permutations of the word BIBLICAL are there?
Try the following:
DIGIT
DEFEATED
MILLION
Permutations of n objects taken r at a time
Example: In how many ways can 7 contestants take the first 3 spots of a pageant?
Permutations of n objects taken r at a time is given by n Pr 
n!
(n  r )!
In how many ways can 100 marathoners take the first five spots?
Simplify the following:
a.
8
P3 4 P2
b.
P5
6 P3
8