Math 2, Unit 8 – Modeling with Probability
Name_____________________
Fundamental Principal of Counting, Permutations
Date____________ Period ____
& Combinations
------------------------------------------------------------------------------------------------------The Fundamental Principle of Counting
How many different outfits could you put together using 2 sweaters, 4 pairs of pants, and
2 pairs of shoes?
The Fundamental Principle of Counting says: Suppose there are ______ ways of choosing one
item, and ______ ways of choosing a second item, and ______ ways of choosing a third item,
and so on. Then the total number of possible outcomes is ____________________.
The probability of an event is: P(Event) =
Ex 1) Suppose a license plate can have any three letters followed by any four digits.
a) How many different license plates are possible?
b) How many license plates are possible that have no repeated letters or digits?
c) What is the probability that a randomly selected license plate has no repeated letters
or digits?
Permutations
Ex. 2) I have five books I want to arrange (in a particular order) on a shelf.
a) How many different ways can I arrange them?
b) What if I only want to arrange 3 of my 5 books on a shelf? How many ways can I do this?
Unit 8: FPC & Permutations & Combination
With Permutations
_____________
_____________
Ex 3: Seven flute players are performing in an ensemble.
a) The names of all seven players are listed in the program in random order. How many
different ways could the players' names be listed (i.e., arranged) in the program?
b) How many different ways could the players' names be listed in alphabetical order by last
name?
c) If the players' names are listed in the program in random order, what is the probability that
the names happen to be in alphabetical order?
d) After the performance, the players are backstage. There is a bench with only room for four
to sit. How many possible arrangements are there for four of the seven players to sit on the
bench?
Permutations vs. Combinations (Electing Officers vs. Forming a Committee)
Ex. 1) We want to elect three officers from our club of 25 people. The first person elected will
be the President, the second person elected will be the Vice President, and the third person
elected will be the Treasurer. How many different “arrangements” of officers can we have?
Note:
Unit 8: FPC & Permutations & Combination
Ex. 2) We want to form a 3-person committee (i.e., no officers) from our club of 25 people.
How many committees can we form?
Note:
When you're counting how many ways there are to ____________ some number of items,
___________ _____________; that's a _____________________.
When you're counting how many ways there are to simply ________ some number of items,
__________ does _______ ____________; that's a ___________________.
Ex. 3) The Debate Club wants to elect four officers (Pres, VP, Sec, and Treas), from its
membership of 30 people. How many different ways could the Debate Club elect its officers?
Ex. 4) The Debate Club wants to create a 4-person committee (i.e., no officers) from its
membership of 30 people. How many different committees are possible?
Combinations with Restrictions
Ex. 5) The Young Republicans Club consists of 7 seniors, 9 juniors, and 5 sophomores. They
want to form a Planning Committee (i.e., without officers) to plan their spring social. The
Planning Committee will consist of 4 members.
a) How many different 4-member committees are possible?
b) How many committees are possible that consist of all sophomores?
Unit 8: FPC & Permutations & Combination
c) How many different committees could be formed if the club's president must be one of the
members?
d) How many different committees could be formed if the committee must contain exactly two
seniors and two juniors?
PRACTICE: FPC & Permutations!!!!!
You must show work for each problem.
1. a) How many possible seven-digit phone numbers are there?
b) How many seven-digit phone numbers are there that begin with the prefix 772?
c) How many seven-digit phone numbers are there that begin with the prefix 772 and none
of the last four digits repeat?
d) What is the probability that a randomly selected phone number with the prefix 772, has
none of its last four digits repeating?
2. Give the number of possible arrangements or selections for each situation.
a) Arrangements of six poetry books on a shelf
b) Arrangements of seven students seated in the front row of a classroom
Unit 8: FPC & Permutations & Combination
c) License plates with two letters followed by four digits.
d) License plates with two letters followed by four digits, or four digits followed by two
letters.
e) Outfits made up of a shirt, a pair of slacks, and a sweater, selected from five shirts, four
pairs of slacks, and three sweaters.
f) Restaurant meals formed by selecting an appetizer, a salad, a main course, and a dessert
from five choices of appetizer, three choices of salad, six choices of main course, and
four choices of dessert.
g) Seven-digit telephone numbers, if the first digit cannot be zero.
3. In Ms. Scarpino’s math class, there are six desks in each row. On the first day of the
semester, she tells her students that they may sit anywhere they want, but that they must sit
in the same row every day.
a) If the first row is completely filled, in how many different ways can the students who
have chosen to sit there be seated?
b) What is the probability that the students in the front row will be seated in alphabetical
order by their first name?
c) What is the probability that among the students in the front row, the tallest student will
sit in the chair farthest to the right?
Unit 8: FPC & Permutations & Combination
d) On April Fool’s Day, the students came to class and found that two of the desks in the
front row were missing. In how many ways could the remaining desks be chosen by the
students who usually sit in the front row?
e) On April Fool’s Day, what is the probability that Ricardo, one of the students who usually
sits in the front row, was able to get a seat in this row?
PRACTICE: Combinations!!!! Evaluate each expression.
1. a)
6C3
= ______
b)
8P7
= ______
c)
24C3
= ______
d)
15P3
= ______
2. Find the number of ways of making each choice.
h) Selecting a 4-member committee from a 20-member club.
i) Selecting a 4-member committee from a 20-member club if the president of the club
must be on the committee.
j) Selecting a 4-member committee from a 20-member club if the president of the club
cannot be on the committee but the treasurer must be on the committee.
k) Selecting a 4-member committee from a 20-member club if there are 12 women and 8
men in the club and the committee must include 2 men and 2 women.
l) Selecting three days out of a week.
Unit 8: FPC & Permutations & Combination
m) Selecting three days out of a week if exactly two of them must be weekdays.
3. The Debate Club wants to create a 4-person committee (i.e., no officers) from its
membership of 30 people.
a) How many different committees are possible?
b) Carlos is a member of the Debate Club. How many different committees are possible
that have Carlos as a committee member?
c) What's the probability that Carlos will be on a committee consisting of
randomly selected members?
4. There are 10 fourth-graders, 12 fifth-graders, and 8 sixth-graders in a Girl Scout troop.
Mrs. Sullivan, the troop leader, needs five girls to serve on the troop’s camping committee. To
make the selection fair, she lets the girls draw names out of a hat to fill the five places on the
committee.
f) How many different committees are possible?
g) What is the probability that Lisa, one of the sixth-grade scouts, will be on the
committee?
h) What is the probability that Lisa and her best friend Naomi will both be on the
committee?
i) What is the probability that all the committee members will be fifth-graders?
j) What is the probability that the committee will be made up of 2 fourth-graders, 2 fifthgraders, and 1 sixth-grader?