Transcript Unit 4 Permutation and Combination

UNIT 4: Applications of Probability
UNIT QUESTION: How do you
calculate the probability of an event?
Today’s Question:
What is the difference between a
permutation and a combination?
Use an appropriate method to find the number of
outcomes in each of the following situations:
1. Your school cafeteria offers chicken or tuna sandwiches; chips
or fruit; and milk, apple juice, or orange juice. If you purchase
one sandwich, one side item and one drink, how many different
lunches can you choose? There are 12 possible lunches.
Sandwich(2)
Side Item(2)
chips
chicken
fruit
chips
tuna
fruit
Drink(3)
Outcomes
apple juice
orange juice
milk
apple juice
orange juice
milk
chicken, chips, apple
chickn, chips, orange
chicken, chips, milk
chicken, fruit, apple
chicken, fruit, orange
chicken, fruit, milk
apple juice
orange juice
milk
apple juice
orange juice
milk
tuna, chips, apple
tuna, chips, orange
tuna, chips, milk
tuna, fruit, apple
tuna, fruit, orange
tuna, fruit, milk
Multiplication Counting Principle
• Ex 2: Georgia, the standard license
plate has 3 letters followed by 4 digits.
How many standard license plates are
available in Georgia?
26
* 26
* 26 * 10 * 10 * 10 * 10
175,760,000
Multiplication Counting Principle
• Ex. 3: At Harrison, the locks on the lockers
have numbers 0-49. Each combination
uses 3 numbers. How many locker
combinations are possible (assuming you
can repeat numbers)?
50
•
* 50
* 50
125,000
Ex 4: Your iPod players can vary the order in which songs are
played. Your iPod currently only contains 8 songs (if you’re a
loser). Find the number of orders in which the songs can be
played.
There are 40,320 possible song orders.
1st Song
2nd
3rd
4th
5th
6th
7th
8th
Outcomes
You don’t want to play the same song twice!
8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
The solution in this example involves the product of all the
integers from n to one (n is representing the starting value).
The product of all positive integers less than or equal to a
number is a factorial.
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
Ex 5: The student council of 15 members must choose a
president, a vice president, a secretary, and a treasurer.
There are 32,760 permutations for choosing the class officers.
President
Vice
Secretary
Treasurer
Outcomes
In this situation it makes more sense to use the
Fundamental Counting Principle.
15 • 14 • 13 • 12 =
32,760
Let’s say the student council members’ names were: Hunter,
Bethany, Justin, Madison, Kelsey, Mimi, Taylor, Grace, Meghan,
Tori, Alex, Paul, Whitney, Randi, and Dalton. If Hunter,
Maighan, Whitney, and Alex are elected, would the order in
which they are chosen matter?
President
Is Hunter
Vice President
Meghan
Secretary
Whitney
Treasurer
Alex
the same as…
Whitney
Hunter
Alex
Meghan?
Although the same individual students are listed in each example
above, the listings are not the same. Each listing indicates a different
student holding each office. Therefore we must conclude that the
order in which they are chosen matters.
Permutation
When deciding who goes 1st, 2nd, etc., order is important.
A permutation is an arrangement or listing of objects in a specific
order.
The order of the arrangement is very important!!
n!
nPr =
(n  r )!
*Note if n = r then nPr = n!
Permutation
Notation
The number of arrangements of “n” objects, taken “r” at a time.
Permutations
Simplify each expression.
a. 12P2 12 • 11 = 132
b. 10P4 10 • 9 • 8 • 7 = 5,040
c. At a school science fair, ribbons are given for first,
second, third, and fourth place, There are 20 exhibits in
the fair. How many different arrangements of four
winning exhibits are possible?
= 20P4 = 20 • 19 • 18 • 17 = 116,280
Permutations
1.Bugs Bunny, King Tut, Mickey Mouse and
Daffy Duck are going to the movies (they
are best friends). How many different ways
can they sit in seats A, B, C, and D below?
A
B
C
D
24
2. Coach is picking a captain and co-captain
from 15 seniors. How many possibilities
does he have if they are all equally likely?
210
Combinations
• A selection of objects in which order
is not important.
• Example – 8 people pair up to do an
assignment. How many different
pairs are there?
Combinations
AB AC AD AE AF AG AH
BA BC
BD BE BF
BG BH
CA CB
CD CE
CG CH
CF
DA DB DC DE DF DG DH
EA EB EC
ED EF
EG EH
FA FB
FD FE
FG FH
FC
GA GB GC GD GE GF GH
HA HB HC HD HE HF HG
Combinations
• The number of r-combinations of a
set with n elements,
• where n is a positive integer and
• r is an integer with 0 <= r <= n,
• i.e. the number of combinations of r
objects from n unlike objects is
n!
n Cr 
r !  n  r !
Example 1
• How many different ways are there to
select two class representatives from a
class of 20 students?
• The number of such combinations is:
20!
 190
20 C2 
2!18!
Example 2
From a class of 24, the teacher is
randomly selecting 3 to help Mr.
Griggers with a project. How many
combinations are possible?
24!
 2024
24 C3 
3! 21!
You try:
For your school pictures, you can
choose 4 backgrounds from a list of 10.
How many combinations of backdrops
are possible?
10!
 210
10 C4 
4! 6!
Clarification on Combinations and
Permutations
• "My fruit salad is a combination of
apples, grapes and bananas"
We don't care what order the fruits are in,
they could also be "bananas, grapes and
apples" or "grapes, apples and bananas",
its the same fruit salad.
Clarification on Combinations and
Permutations
• "The combination to the safe is 472".
Now we do care about the order. "724"
would not work, nor would "247". It has to
be exactly 4-7-2.
To sum it up…
• If the order doesn't matter, then it is a
Combination.
• If the order does matter, then it is a
Permutation.
A Permutation is an ordered Combination.
Classwork
• p. 178, #1-8
Homework
• p. 178-180, #9-15 all, 17-31 odd, 35-43
odd