AP Stats
BW
10/14 or 15
Suppose that the numbers of unnecessary procedures
recommended by five doctors in a 1-month period are
given by the set {2, 2, 8, 20, 33}.
If it is discovered that the fifth doctor recommended
an additional 25 unnecessary procedures, how will
the median and mean be affected?
Section 3.4 – Counting Principles
SWBAT:
Use Permutations and Combinations in finding probabilities.
Source: www.edu-resource.com
Counting Principles
In Section 3.1, we learned the Fundamental
Counting Principle is used to find the number of
ways two or more events can occur in sequence.
In this section we will study several other
techniques for counting the number of ways an
event can occur.
Source: www.slideshare.net
Permutations:
nPr =
The number of permutations of n distinct
objects taken r at a time is
,
where r < n
EX: Four people went to the movies and there are four open seats
next to each other in the middle of the movie theatre. How
many ways can they arrange themselves?
Using Fundamental Counting Principle:
___ _ * _____ * _____ * ____
Seat1 Seat2 Seat3 Seat4
Using Permutations:
nPr = = 4P4 =
4 * 3 * 2 * 1 = 24
= 24 ways
Permutations, Cont’d
EX: How many arrangements are there of the word ANGLE?
Using Fundamental Counting Principle:
5! = 5 * 4 * 3 * 2 *1 = 120
Using Permutations:
5P5 =
EX: What if four people go to a movie, but there are only two open seats next
to each other? How many distinct ways can these four people arrange
themselves in the two seats?
Using Fundamental Counting Principle:
4 * 3 = 12
Using Permutations:
4P2 =
You Try…. Use Permutations to find…
EX: Six people are running for four different offices on the
school board. In how many different ways can those offices
be filled?
6P4
= 360 ways
EX: Twelve horses are running in a race. How many different
ways can 1st, 2nd, and 3rd place be awarded?
12P3
= 1,320 ways
EX: How many 3-letter words can be formed from the “word”
TEXAS if each letter is used only once in a “word”?
5P3
= 60 ways
Let’s try a trickier one…
a)
In how many ways can the letters A, B, C, D and E be
arranged in a row?
5P5
b)
= 120 ways
In how many of these arrangements is D always the fist?
1 * 4 * 3 * 2 * 1 = 24
4P4
= 24 ways
Distinguishable Permutations:
Order a group of n objects in which some objects are the same
EX: How many ways can I arrange the letters AAAABBC?
You Try….
EX: How many different ways can I arrange the letters in the
words below?
a) APPLE
b) DEGREE
c) DIVIDED
You Try….
EX: A building contractor is planning to develop a subdivision. The
subdivision is to consist of 6 one-story houses, 4 two-story
houses, and 2 split-level houses. In how many distinguishable
ways can the houses be arranged?
Combinations:
A selection of r objects from a group of n
objects without regard to order
nCr =
EX: I select two people from class to go to Hawaii.
I select 3 toppings on a pizza.
Vs.
Choosing a 2 digit password…. Order would matter, so it would be
a permutation
Combinations, cont’d
EX: You have five choices of sandwich fillings. How many different
sandwiches could you make by choosing three of the five
fillings?
5C3 =
EX: Katie is going to adopt kitten from a litter of eleven. How
many ways can she choose a group of 3 kittens?
11C3 =
Let’s try a trickier one…
There are fourteen juniors and twenty-three seniors in the Service
Club. The club is to send four representatives to the State
Conference.
a) How many ways are there to select a group of four students to
attend the conference?
37C4 =
66,045 ways
b) If members of the club decide to sent two juniors and two
seniors, how many different groupings are possible?
14C2 * 23C2 =
23,023 ways
You try…..
Tell whether each situation represents a
permutation or a combination?
a. C
a. A coach chooses a team of 6 players from 12.
b. Ten people are in a line to buy tickets.
b. P
c. A teacher selects a committee of 4 students
from 25 students.
c. C
d. The different groups of three vegetables you
could choose from six different vegetables.
d. C
e. Different orders you can play 4 DVDs.
f. Different groups of class officers students can
elect from a class of 25 students.
e. P
f. P
One more…
A student advisory board consists of 17 members. Three
members serve as the board’s chair, secretary, and webmaster.
Each member is equally likely to serve any of the positions. What
is the probability of selecting at random the three members that
hold each position.
a) Find the number of ways the three positions can be filled.
17P3 =
4080 ways
b) Find the probability of correctly selecting the three members
OK…really…One more…
Sue Bartling loves to read mystery books and car-repair manuals. On a visit to
the library, Sue finds 9 new mystery books and 3 car-repair manuals. She
borrows 4 of these books. Find the number of different sets of 4 books Sue can
borrow if:
a) All are mystery books
9C4 =
126 different sets
b) Exactly 2 are mystery books
9C2 * 3C2 =
c)
108 different sets
Only one is a mystery book
9C1 * 3C3 =
9 ways
NOTE: These have to
add up to 4
You try…..
Find the probability of being dealt five diamonds from a standard deck of playing
cards.
-All 5 together, order does not matter (group of 5)
-# of ways of choosing 5 diamonds:
-# of possible 5 card hands:
- Probability:
13C5
/
13C5
52C5
52C5
= 0.000495
HOMEWORK:
Worksheets:
3.4 Practice: Combinations & Permutations
3.4 – Applications of the Counting Principle