Transcript Permutations - Salesianum School

5.4 Counting Methods
Objectives:
By the end of this section, I will be
able to…
1)
Apply the Multiplication Rule for Counting to
solve certain counting problems.
2)
Use permutations and combinations to solve
certain counting problems.
3)
Compute probabilities using combinations.
SAT problems
In
how many ways can you
pick….
Counting Rule
An easy way to find total
possibilities or the
SAMPLE SPACE
Counting Rule
MULTIPLY
LICENSE PLATES
 In
Delaware, license plates have seven
characters. How many possibilities are there
total?
36·36·36·36·36·36·36=
26·25·24·23·22·21·20=
 What if you could only use numbers?
E letters?
10
 What if you could 7.836
only3,315,312,000
use
10·10·10·10·10·10·10=
 What if you can’t repeat letters?
26·26·26·26·26·26·26=
 What if you do not have
any restrictions at
10,000,000
8,031,810,176
all?
PASSWORDS
 Most
internet sites force you to have a
password.
 If
one site requires 8 characters, letters
only, how many possibilities are there?
26·26·26·26·26·26·26·26=
2.088 E 11
CELL PHONES
 How
many phone numbers are possible in
Delaware?
 Are there enough possibilities if everyone
had their own cell phone? (There are about
850,000 people living in Delaware)
(302)…-….
1·1·1·8·10·10·10·10·10·10=
8,000,000
PERMUTATIONS
Any arrangement of distinct
objects in a particular order.
Factorial Notation
1)
2)
3! = 3 x 2 x 1
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
ON YOUR CALCULATOR
MATH – PRB – !
Check this out!
 PERMUTATIONS
THE MOVIE
PERMUTATION FORMULA
P

n r
n!
(n  r )!
n = total
r = how many you are selecting
Rankings
 In
how many different ways can you
arrange the following beautiful ladies?
Sports
In
how many ways can
teams finish in the NFC
East?
Starting Line UP
In
how many ways can Coach
Dinardo choose his starting
offense from a total of 32
players?
Combination
An arrangement of objects
where order is NOT important.
Combination Formula
The
number of combinations of
n total things taken r at a time
is denoted as
C

n
r
n!
r !( n  r ) !
 Angel
always dresses well. He is
deciding which 2 shirts to wear to
school this week out of his 13 clean
shirts.
 In how many ways can Angel choose
which two shirts to wear?
 Serpe
is cleaning his room and actually
putting his collection of baseball cards
away. He usually puts 4 new cards on
display each month.
 In how many ways can he choose the 4
cards from a possible 32 cards?
Permutation vs. Combination
Both
questions ask “in how
many ways…”
Permutations need ORDER

Ranking
Combinations

More random
DON’T
1)
2)
The senior class is going to take the
top four students (by GPA) to Jamaica
for a little R&R. In how many ways can
they be chosen?
What if they pick the four students
randomly from a hat. In how many
ways can they be chosen?
Poker!
 How
many different poker hands
consisting of five cards can be dealt
from a deck of 52 cards?
 What
is the probability of being dealt
a royal flush in five card poker?
PASCAL’s TRIANGLE

It is a mathematical relationship between
numbers that can be used for computing the
number of possible combinations of n things
taken r at a time.

Use Pascal’s Triangle to find the following.
a.
C
5 2
b.
C
5 3
c.
C
6 6
d.
C
7 0
1 6 15 20
15 6 1