HSS-CP.B.9 STUDENT NOTES & PRACTICE WS #1/#2
1
Counting Principles
In calculating probabilities we often need to determine either the total number of possible outcomes or the
total number of successful outcomes. In many cases the outcomes are easy things to count… red aces, blue
marbles, coin flips, etc.. but sometimes determining these values can be a little more complex.
For example, Jimmy remembers the first 3 digits of Randy’s phone number but not the last 4. How many
different possible numbers are there?
248 - ____ ____ ____ ____
There are 10 different digits 0 – 9 for each spot so there would be 10 • 10 • 10 • 10 = 10,000 different phone
numbers. So Jimmy has a 1 / 10,000 chance of getting the number correct….. Good Luck Jimmy!!!
As mentioned in an earlier objective, the fundamental counting principle is a powerful tool to helping us with
problems like this.
FUNDAMENTAL COUNTING PRINCIPLE
If you can choose one item from a group of M items and a second item from a group of N items, then the total
number of two items choices is M • N.
The FCP allows for the use of the same item to be used more than once. For example, in the telephone
example above 248 – 1212 is a perfectly fine result. It is for this reason that we multiply 10 • 10 • 10 • 10
because the numbers 0 – 9 can be used for all four spots if necessary.
This is not always the case, sometimes when we use a number or an item it cannot be used again, for example,
if there are four people and we want to know how many different ways that they could finish a race, a person
cannot come in 1st and in 4th. Thus 4 • 4 • 4 • 4 doesn’t make sense – once a person places they cannot be
used again.
Permutation (Order Matters)
Let’s try a little different problem. Four bands are going to perform at the Friday Night Jam Session. How
many different ways could the schedule for the night be arranged?
The way I make sense of this is 4 different bands could go first, since one has been picked 3 different bands
could go in the next place, then 2 and then finally 1…. 4 • 3 • 2 • 1 = 24 different ways. In mathematics when
we multiply n (n – 1) (n – 2) …. We call that a FACTORINAL AND IT IS WRITTEN,
6! = 6 • 5 • 4 • 3 • 2 • 1 = 720
3! = 3 • 2 • 1 = 6
1! = 1
n!
and by definition 0! = 1
The band problem has a few important characteristics that our earlier fundamental counting principle
problems didn’t have:
1. NO ITEM IS USED MORE THAN ONCE
2. THE ORDER OF THE ARRANGEMENT MAKES A DIFFERENCE.
When these two conditions are involved in the arrangement it is called a PERMUTATION.
HSS-CP.B.9 STUDENT NOTES & PRACTICE WS #1/#2
2
When does ORDER MATTER? This is an important question!! Order matters when the sequence means
something. Jack is runner J, Sally is runner S and Randy is runner R so if they finish the race SJR the order
matters because Sally was 1st, Jack came in 2nd, and Randy came in 3rd. If they have finished RSJ, they would
receive different prizes and awards. In contrast, if these three runners had been selected to get a $10 gift
certificate for finishing in the top three, it wouldn’t have matter if they finished SJR or RSJ – they would all get
the same prize and the order wouldn’t matter at all. Order matters when the value or meaning of the location
is somehow different from the other locations.
When order matters 123, 132, 213, 231, 312 and 321 are all different possible arrangements for the situation.
If order didn’t matter 123, 132, 213, 231, 312 and 321 are all the same arrangement for the situation.
In the below examples, note that no items can be used more than once and that the order matters. Thus
these are examples of PERMUTATIONS.
If you have 4 books. How many different ways can
you arrange these books assuming that the order
matters to you?
?
4
?
x
3
?
x
2
There are 10 people in the race. How many different
ways could they finish 1st, 2nd and 3rd ?
?
?
x
1
?
10
= 24
x
?
9
x
8
= 720
We see a nice pattern forming through this intuitive approach to solving the problem. Each time one less
option is available because an item cannot be used more than once. Also if there are more items than things
that we are choosing then we only do that many locations.
The above method for calculating permutations is a very intuitive method (but very effective). There is a
powerful formula that summarizes exactly what we have been doing above.
n!
P
=
n r
(n − r )!
10 P3 =
10!
10!
=
(10 − 3)! 7!
10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1
7 • 6 • 5 • 4 • 3 • 2 •1
10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 •1
7 • 6 • 5 • 4 • 3 • 2 •1
10 • 9 • 8 = 720
n is the total number of elements and r is the
number of items being chosen. So in our
second example above n = 10 and r = 3, we
read this notation to be the Permutation of n
choose r or in this specific question, we would
read it the permutation of 10 choose 3.
?
10
?
x
9
?
x
8
= 720
HSS-CP.B.9 STUDENT NOTES & PRACTICE WS #1/#2
3
Combination (Order DOESN’T Matters)
Let me introduce another type of problem. A teacher wants to pick 2 students to get the textbooks from the
other room. If there are 5 students, how many groups can be formed? What is different about this question
from the Permutation questions earlier? ORDER DOESN’T MATTER!! There is no difference to being chosen
1st or 2nd. In this situation, Mike and then Joe is the same as picking Joe and then Mike… the pair of students
selected for the group is still Mike and Joe This is known as a COMBINATION.
1. NO ITEM IS USED MORE THAN ONCE
2. THE ORDER OF THE ARRANGEMENT MAKES A NO DIFFERENCE.
(Picking 123 is the same as picking 231 or 321 or 213… etc…. there are always fewer combinations than
permutations because combinations divide out the ‘repeated equivalent’ groups.)
We will introduce formulas soon but for now a combination is determined by finding the permutations (all
arrangements) and then dividing out the duplication by dividing by r!, where r is the grouping size.
From 7 students a committee of 3 is being formed.
How many different groups of 3 could be formed?
?
7
?
x
?
6
x
210
3!
5 books lay on a table. Jeff needs to pick two to read
for an English assignment. How many different
groups of 2 could be formed?
5
?
5
= 210
210
=
6
?
x
20
= 35
2!
4
= 20
20
=
2
= 10
We notice that this is the same process as determining a permutation EXCEPT WE DIVIDE OUT THE
DUPLICATION. This helps us to alter the formula for permutations into one that will calculate the
combinations.
n!
C
=
n r
(n − r )!r !
7!
C
=
= 35
7 3
(7 − 3)!3!
n is the total number of elements and r is the
number of items being chosen. Notice the
slight difference to the permutation formula,
we are also dividing by r!, to take out all the
duplication.
?
7
?
x
?
6
x
210
We read this notation to be the Combination of n
choose r or in this specific question, we would read it
the combination of 7 choose 3.
3!
5
210
=
6
= 35
= 210
HSS-CP.B.9 STUDENT NOTES & PRACTICE WS #1/#2
4
Determine if the following situations are (F)undamental Counting, (P)ermutations or (C)ombinations.
a) In how many different ways can
3 people finish a race?
b) How many 4 digit passcodes are
possible?
PERMUTATION
FUNDAMENTAL
d) 5 questions on a multiple choice
quiz and each question has 4
possible answers. How many
different possible ways could you
answer the quiz?
e) 20 old video games are in a bin.
The sign says pick any 4 for $5.
How many different groups of 4
are possible?
c) In how many different ways can
a 2-person committee be selected
from 10 people?
COMBINATION
f) 4 bands are performing on the
program. How many different
ways could they be arranged?
PERMUTATION
COMBINATION
FUNDAMENTAL
Now solve the above questions.
a)
3P3
b)
(10)(10)(10)(10) = 10,000
=6
d)
c)
e)
(4)(4)(4)(4)(4) = 1024
10P2
= 45
4P4
= 24
f)
20C4
= 4845
Permutations & Combinations with a Calculator
Most graphing calculators have these relationships pre-programed into the calculator. Let me show you were
these special probability operations can be found.
EXAMPLE – TI-84 CALCULATOR
The probability relationships are
found under the MATH button on
your calculator.
Use the Right Arrow to
Scroll to PRB (Probability)
They are all here….
To use the permutation or combination formulas you must (1) first enter the number of elements, n, and then
(2) choose which operation you want, permutation or combination, and then (3) finally enter the number of
items being chosen, r. This will calculate the permutation or combination completely.
HSS-CP.B.9 WORKSHEET #1
Name: _____________________________ Period ______
1
1. Read the questions from #2 and then determine their type of question.
(F) Fundamental Counting Principle, (P) Permutation or (C) Combination.
a) F or P or C
b) F or P or C
c) F or P or C
d) F or P or C
e) F or P or C
f) F or P or C
g) F or P or C
h) F or P or C
i) F or P or C
j) F or P or C
k) F or P or C
l) F or P or C
2. Calculate the total number of possible outcomes.
a) In how many different ways can
five students be seated in a row of
five desks?
_________
b) From a wardrobe of 10 sweaters, 8c) In how many different ways can a
jeans, and 4 pairs of shoes, how many3-person committee be selected from
different outfits consisting of a
6 people?
sweater, jeans, and a pair of shoes
can a student choose?
_________
_________
d) In a random drawing, three names e) In how many different ways can f) In how many different ways can the
first, second, and third prize winners letters in the word FACTOR be
are chosen to win the same
computer. If 30 names are entered inbe chosen in a random drawing if 50 scrambled?
the contest, how many different sets people entered a contest?
of winners can be chosen?
_________
_________
_________
g) A student, taking a true-false test, h) The telephone numbers in a
i) How many ways can you choose
randomly guesses at every one of the certain area of the city all start with two jellybeans from a bag of 15
10 answers. How many different sets865, and end with four more digits. (order matters & no-replacement)?
of answers could be produced?
How many possible phone numbers
are there in this area of the city?
_________
j) How many groups of 3 toys can a
child choose to take on a vacation
from a toy box containing 11 toys?
_________
_________
k) There are 24 students in the class
and 5 desks in the front row. How
many different ways could the front
row be filled out?
_________
_________
l) A video store has 27 new release
movie posters. How many ways can
the manager choose a group of 4 for
a prize?
_________
HSS-CP.B.9 WORKSHEET #1
2
3. Calculate the total number of possible outcomes.
a) In how many different ways can the letters in the
word PRIZE be scrambled?
b) Each morning a boy chooses his own type of cereal
and eggs for breakfast. If he has three types of cereal
and four types of eggs from which to choose, how many
different breakfasts can he choose?
c) A committee of 3 students is to be selected from a d) At a restaurant there are four types of beverages and
group of 10 to be on a committee to plan a school trip. six types of sandwiches. How many different orders
How many different combinations can be selected?
consisting of one beverage and one sandwich are there?
e) A map is to be colored using exactly four colors. If f) Each of the entries in a contest is marked with a twoseven different colors are available, in how many groupscharacter code, a letter followed by one of the digits
from 0 to 9. How many different character codes are
of four colors can be chosen?
possible?
g) A restaurant menu features 5 different appetizers andh) 10 Standbys are hoping to get on a plane. If there is
room for only 3 to make it on the flight. How many
3 different main dishes. A diner decides to order 2
appetizers and 1 main dish. How many such different different groups of 3 could be formed?
orders can he make?
i) A game contains 15 cards, numbered 1 to 15. How
many different ways can you deal out 5 cards if the
order they are dealt matters to the game?
j) Jeff has 5 T-Shirts, 3 pants, and 4 pairs of shoes. How
many different outfits could he make with these items?
k) A police lineup of 5 suspects is being created. How
many different ways can you arrange these 5 people?
l) There are 20 people in a raffle. First pick gets $50,
second gets $25 and third gets $10. How many different
arrangements of winners could there be?
I
I
I
I
HSS-CP.B.9 WORKSHEET
#1
it
Nome: #4-
period
1. Read the questions from #2 and then determt^
(F) Fundamental Counting Principle, (P) Permutation
a)F or@or c
b)F or P or
C
c)F or P or@
d)F or P or
C
e)F or@or
C
f)
F or P or
C
8@or P or
C
h)F or P or
C
i)
F or@or
C
k)F or@or c
l)
F or P or
C
j)
2.
or (C) Combination.
F or P or
C
Calculate the tota! number of possible outcomes.
a) ln how many different ways can b) From a wardrobe of L0 sweaters, I c) ln how many different ways can a
five students be seated in a row of jeans, and 4 pairs of shoes, how man 3-person committee be selected fror
five desks?
different outfits consisting of a
6 people?
sweater, jeans, and a pair of shoes
C?_
can a student choose?
!
6 3
5
/,a
l7-o
d) ln a random drawing, three name: e) ln how many different ways can f) ln how many different ways can th
are chosen to win the
first, second, and third prize winners letters in the word FACTOR be
computer. lf 30 names are entered i be chosen in a random drawing if 50 scrambled?
the contest, how many different sets people entered a contest?
of winners can be
same
\--
chosen?
g)A student, taking
a true-false test,
randomly Buesses at every one of thr
L0 answers. How many different set
of answers could be produced?
?
gb ,
ll1,(m
a
h)The telephone numbers in
i) How many ways can you choose
certain area of the city all start with two jellybeans from a bag of L5
865, and end with four more digits. (order matters & no-replacement)?
How many possible phone numbers
are there in this area of the
city?
JN
T
ttL
alo
Prq
j) How many groups of 3 toys can a k)There are24 students in the class l) A video store has 27 new release
child choose to take on a vacation and 5 desks in the front row. How movie posters. How many ways can
from a toy box containing 11 toys? many different ways could the front the manager choose a group of 4 for
row be filled
zqfs
out?
a prize?
D,uo,,rtro
H SS.CP,
3.
8.9 WO RKSH EET # 7
Calculate the
2
total number of possible outcomes.
a) ln how many different ways can the letters in the
word PRIZE be scrambled?
?.
=@
5 5
b) Each morning a boy chooses his own type of cereal
and eggs for breakfast. lf he has three types of cereal
and four types of eggs from which to choose, how man'
different breakfasts can he choose?
\-s-
c) A committee of 3 students is to be selected from a
d) At a restaurant there are four types of beverages anc
group of L0 to be on a committee to plan a school trip. six types of sandwiches. How many different orders
How many different combinations can be selected?
consisting of one beverage and one sandwich are there
Mof -
@@
to be colored using exactly four colors. lf f) Each of the entries in a contest is marked with a twoseven different colors are available, in how many group character code, a letter followed by one of the digits
of four colors can be chosen?
from 0 to 9. How many different character codes are
e) A map is
*r=@Possibre?
@
g) A restaurant menu features 5
different appetizers an h) 10 Standbys are hoping to get on a plane. lf there
3 different main dishes. A diner decides to order 2
room for only 3 to make it on the flight. How many
appetizers and 1 main dish. How many such different different groups of 3 could be formed?
orders can he
make?
S ?=@
i) A game contains 15 cards, numbered i. to 15. How
many different ways can you deal out 5 cards if the
order they are dealt matters to the game?
rsPs
is
j) Jeff has 5 T-Shirts, 3 pants, and 4 pairs of shoes. How
many different outfits could he make with these items?
:@@
k) A police lineup of 5 suspects is being
created. How
many different ways can you arrange these 5 people?
l) There are 20 people in a raffle. First pick gets 950,
second gets S25 and third gets $t0. How many differer
arrangements of winners could there be?
r
5'5 / l2o