Transcript Counting Rule

How many ways can we arrange 3 objects A, B, and C:
•Using just two 6
How many ways can we arrange 4 objects, A, B, C, & D:
•Using only two 12
•Using only three 24
Keep this. We will get back to this later today!
Math I
UNIT QUESTION: How do we
determine the number of options if
order matters?
Standard: MM1D1.b
Today’s Question:
How can we find the number of ways
to make a 9 team batting order out of
20 people?
Standard: MM1D1.b.
Factorial (!)
The product of counting numbers beginning at n and
counting backward to 1 is written n! and it’s called n
factorial.
factorial.
EXAMPLE with Songs
‘eight factorial’
8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
Factorial
Simplify each expression.
a. 4! 4 • 3 • 2 • 1 = 24
b. 6! 6 • 5 • 4 • 3 • 2 • 1 = 720
c. For the 8th grade field events there are five teams: Red,
Orange, Blue, Green, and Yellow. Each team chooses a
runner for lanes one through 5. Find the number of ways
to arrange the runners. = 5! = 5 • 4 • 3 • 2 • 1 = 120
Factorial
3! = 6
2! = 2
1! = 1
0! = 1
Definition: 0! Equals 1
The bowling league has 8
players. How many ways are
there to line up the bowlers?
(Answer: 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320)
How can we represent this using factorials? 8!
This works well if we are using all of our objects. What if
we only use a part of our objects?
Try to state our data from the warm-up using a ratio of
factorials:
How many ways can we arrange 3 objects A, B, and C:
•Using just two – 6 ways
•Hint:
3 objects, 2 at a time
How many ways can we arrange 4 objects, A, B, C, & D:
•Using only two – 12 ways
•Hint:
4 objects, 2 at a time
•Using only three – 24 ways
•Hint: 4 objects, 3 at a time
How many ways can we pick three
people from a group of 12 if order
matters?
(Answer: 12 * 11 * 10 = 1320)
How can we represent this by using factorials?
(Hint: 12 objects taken 3 at a time)
12! / 9!
Permutations
 The arrangement of elements in a distinct order is
called a permutation. The number of permutations
on n objects, taken r at a time is:
 The previous example would be
12
P3 
12 !
12  3 !

12 !
9!
 1320
 You are selecting a 9 person baseball team out of 12
students and making the batting order.
 Does order matter?
 YES! – therefore it is a permutation problem
n
Pr 
n!
n  r 

12 !
12  9 
12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
3 * 2 *1
 79 ,833 , 600
Distinguishable Permutations
 How many distinguishable ways can the letters MOO





be arranged?
3! Is not the right answer, because the two O’s look the
same
Make a tree diagram
The answer is 3
How do we show this as factorial numbers?
3!/2!
Distinguishable Permutations
 How many distinguishable ways can the letters





WOOF be organized?
Again, 4! Is not right answer.
Make a tree diagram
The answer is 12
How do we show this as a factorial?
4!/2!
Distinguishable Permutations
 How many distinguishable ways can 2 similar marbles
and three similar blocks be arranged?
 There are 5 objects. How many ways can 5 objects be
arranged?
 5! is too big because the marbles are not
distinguishable, and the 3 blocks are not
distinguishable.
 Make a tree diagram
Distinguishable Permutations
 The answer is 10
 How do we show this as factorial numbers?
 5!/(3!2!)
 Distinguishable Permutations 
n!
n1 ! n 2 ! n 3 !... n k !
Permutation Review
 The number of unique ways of organizing 6 unique
things is 6! = 720 ways
 The number of unique ways of organizing 6 unique
things in groups of 4 is 6!/2!, or
6
P4 
6!
6  4 !

6!

2!
720
2
 360
 The number of unique ways of organizing 6 things,
but 4 are the same and 2 are the same is:
6!
4!2!
 15
Class Work
Page 344, # 1 – 28 all, and find the
number of unique permutations for
the letters of the following words:
mississippi
armageddon
supercalifragilisticexpialidocious