Transcript Slide 1 - Fort Bend ISD

Counting Principle & Permutations
I. Counting Principle: If there are m ways of doing one thing and
n ways to do another, then there are
m x n ways of doing both.
Example: You have 4 shirts and 3 pairs of pants that all go together.
How many outfits can you create from them? 4 x 3 = 12 outfits.
A) Only works if all choices are independent of each
other. (In the example above all the clothes “match” with
each other so they can be worn in any combination.)
B) If one choice affect the other choice, then the math
changes.
1) Do each choice separately, then add the totals
together.
Counting Principle & Permutations
Examples of dependent choices:
1)
You want to buy a new car. There are only two cars body
styles (hatchback and 4-door) that you want. Each body
style is available in different colors. The hatchback is
available in 6 colors, but the 4-door is only available in 5
colors. Furthermore, each body style is available in 3
model styles, GL, SS, SL. How many combinations are
there?
Hatchback = 6 colors x 3 styles = 18 choices
4-door = 5 colors x 3 styles = 15 choices
Total choices = 18 + 15 = 33 choices.
Counting Principle & Permutations
II. Permutations = an arrangement of objects in a specific order.
A) Writing Permutation symbol & their meanings: n P r
n = the number of objects in the set.
P = the symbol for permutations (order matters)
r = how many objects in the set you are arranging.
n!
Pr=
n  r !
B) Formula for Permutations: n
1) the ! symbol means factorial.
3! = 3 x 2 x 1,
5! = 5 x 4 x 3 x 2 x 1, 0! = 1, etc.
n!
2)
is used if the objects cannot be repeated.
n  r !
Counting Principle & Permutations
Permutations without repetition examples
1) You have 4 books you want to arrange on a shelf. How many
ways can you arrange them?
(have 4 objects = n, want to arrange all 4 of them = r)
4!
4  4 !
=
4 x 3 x 2 x1
1
the bottom = 1 because 0! = 1
there are 24 ways to arrange the books.
2) You have 6 books, but you only have room to put 4 of them
on shelf. How many ways can you arrange them?
(have 6 objects = n, want to arrange only 4 of them = r)
6!
6  4 ! =
6 x5 x 4 x 3x 2 x1
=
2 x1
360 way to arrange 4 books.
Counting Principle & Permutations
III. Permutations with repetition.
A) Since we can use the same object over and over, the
math formula changes.
1) This type of permutation is used for things like the
combination to your locker.
2) Formula: nr
n = number of objects.
r = how many objects you are arranging.
Example: If a bike chain combination lock has 4 places for
numbers and each place has 10 numbers, how many
combinations are there for the lock?
10 numbers to chose from, 4 objects (places) to arrange =
104 = 10,000 different combinations.