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(13 – 1)
The Counting Principle and Permutations
Learning targets: To use the fundamental counting principle to count the
number of ways an event can happen
To use permutations to count the number of ways an event can
happen
The Fundamental Counting Principle
(also known as the multiplication rule for counting)
If a task can be performed in n1 ways, and for each of these a second task
can be performed in n2 ways, and for each of the latter a third task can be
performed in n3 ways, ..., and for each of the latter a kth task can be
performed in nk ways, then the entire sequence of k tasks can be
performed in n1 • n2 • n3 • ... • nk ways.
I do: (ex) A sandwich stand has four kinds of meat, ham, turkey, roast
beef, and chicken. Also they have three kinds of bread, white, wheat,
and rye. How many choices do you have?
White
Ham
Wheet
3 choices with ham
Rye
White
Turkey
Wheet
3 choices with turkey
Rye
White
Wheet
Roast beef
Rye
White
Chicken
3 choices with roast beef
Wheet
Rye
3 choices with chicken
Total 12 choices
4 3  12
We do: (ex) You can choose 3 different flowers out of 7 with the same
price. How many choices you have?
The first choice out of 7: 7 choices
The second choice out of 6: 6 choices
The third choice out of 5: 5 choices
Total choices: (7)(6)(5)=210 choices
You do: (ex) You have 5 T-shirts and 3 pair of pants. How many
choices of outfit do you have?
(13 – 1) continued
Using Permutations
Definition:
A permutation an order of n objects. (Order is the matter)
It is the same as the Fundamental Counting Principle.
Notation:
The number of permutations of r objects taken from a group of n
distinct objects is given by:
n!
n
Pr 
(n  r )!
Note: A symbol ! is called “factorial”.
n! = n (n – 1)(n – 2)(n – 3)…(1)
(ex) 5! = 5 (4) (3) (2) (1) = 120
Remember: 0! = 1
I do (ex) What is the total number of possible 4-letter arrangements of
the letters m, a, t, h, if each letter is used only once in each
arrangement?
1st letter
4 choices
2nd letter
3 choices
3rd letter
2 choices
4th letter
1 choice
Total
4! = 24 arrangements
We do (ex) Number plate
The number plate of a car consists of 3 letter and 4 numbers. How
many ways can they arrange without repetition?
1st letter: 26 choices
2nd letter: 25 choices
26!
26 P3 
(26  3)!
3rd letter: 24 choices
AND
1st number 10 choices
2nd number 9 choices
3rd number 8 choices
4th number 7 choices
Total arrangements:
10 P4 
10!
(10  4)!
You do: (ex) Joleen is on a shopping spree. She buys six tops, three
shorts and 4 pairs of sandals. How many different outfits consisting of a
top, shorts and sandals can she create from her new purchases?
A pizza shop runs a special where you can buy a large pizza with one
cheese, one vegetable, and meat for $9.00. You have a choice of 7
cheeses, 11 vegetables, and t meats. Additionally, you have a choice of
3 crusts and 2 sauces. How many different variations of the pizza
special are possible?
Permutation with repetition (distinguishable permutation):
The number of distinguishable permutations of n objects where one
object is repeated q1 times, another is repeated q2 times, and so on is:
n!
(q1 !)(q2 !)...(qk !)
I do: (ex) Find the number of distinguishable permutations of the letter in
Cincinnati?
Cincinnati has 10 letters. (n)
n!
(q1 !)(q2 !)...(qk !)
i is repeated 3 times (q1)
n is repeated 3 times (q2)
c is repeated 2 times (q3)
=
10!
(3!)(3!)(2!)
We do: (ex) Find the number of distinguishable permutations of the letter
in Mississippi?
Mississippi has 11 letters. (n)
i is repeated 4 times (q1)
s is repeated 4 times (q2)
p is repeated 2 times (q3)
You do: (ex) Find the number of distinguishable permutations of the
letter in Calculus?
Find the number of distinguishable permutations of the letters in the
word.
a) PUPPY
d) CONNECTICUT
b) LETTER
c) MISSOURI
Combinations
Definition:
A combination is a selection of r objects from a group of n objects
where the order is not important.
Notation:
c
n r
and
n!
n cr 
(n  r )! r !
I do: (ex) There are 12 boys and 14 girls in Mrs. Brown’s math
class. Find the number of ways Mrs. Brown can select a team of 3
students from the class to work on a group project. The team is to
consist of 1 girl and 2 boys.
It is a combination question because order, or position, is not
important.
To select 2 boys from 12 is:
n = 12 and r = 2
AND
n!
n cr 
(n  r )! r !
n!
To select 1 girl from 14 is: n n cr 
(n  r )! r !
= 14 and r = 1
Total ways:
We do: (ex) You can select at most 3 items out of 10 items on your pizza.
How many different types of pizza do you have as your choices?
•At most means: 0, 1, 2, OR 3 items
The choice of 0 items:
c 10 c0
n r
The choice of 1 item:
The choice of 2 items:
The choice of 3 items
Total choices:
You add if the situation is OR.
You do: (ex) In how many ways can 7 cards be dealt from a deck of 52
cards if you want 4 red and 3 black cards.
(Hint: how many red cards and
black cards are there? What is n
then?)
<Try this> In how many ways can you obtain all 7 red or all 7 black?
Using Binomial Theorem
Complete the next three row of Pascal's Triangle.
(12 – 2) continued
Where
n
   n cr
r 
I do: (ex) Expand
( x  y)4
We do: (ex) Expand
( x  2 y)4
You do: (ex) Expand
( x  2)
6
(13 – 3)
Probability
Definition:
The probability of an event is a number between 0 and 1 that shows
the chance will happen.
Theoretical probability (probability): When an event A will happen
out of all outcomes happen equally, likely.
Theoretical probability is:
number of outcoms in A
P( A) 
total number of outcoms
I do: (ex) What is the probability to get an odd number when you roll a
die?
A die has 6 faces (1~6)
An odd number are 1, 3, or 5
P (rolling 1) 
1
6
1
P (rolling 3) 
6
1
P (rolling 5) 
6
P(rolling an odd number) 
We do (ex) You shuffle the numbers 1~6 and place them. What is the
probability to get 6 as the 1st number?
Total number to fill 6 places: 6!
1st place
2nd place
3rd
4th
5th
6th
6
1(5!)
P(gettubg 6 as the 1st place) 
6!
We do: (ex) Teresa and Julia are among 10 students who have applied
for a trip to Washington, D.C. Two students from the group will be
selected at random for the trip. What is the probability that Teresa and
Julia will be the 2 students selected?
c
10 c2
2 2
You do (ex) There are 20 cards (from 1 through 20). What is the
probability to get a perfect square is chosen?
<Try this> (1) What is the probability to get a face card out of 52-card
deck if you randomly pick one card.
(2) A math teacher is randomly distributing 15 rulers with centimeter
labels and 10 rulers without centimeter labels. What is the probability
that the first ruler she hands out will have centimeter labels and the
second ruler will not have labels?
<try this> (3) One bag contains 2 green marbles and 4 white marbles,
and a second bag contains 3 green marbles and 1 white marble. If Trent
randomly draws one marble from each bag, what is the probability that
they are both green?
Complement of probability
Definition:
The complement of event is that not in the event
Notation:
A’ is called “complement of event A”
Probability of complement of A is: P(A’) = 1 – P(A)
I do: (ex) Two dice are tossed. What is the probability that the sum is
not 8?
P (sum is not 8) = 1 – P (sum is 8)
Sum is 8: (2 + 6), (3 + 5), (4 + 4), (5 + 3), (6 + 2)
total 5 ways
Each die has 6 faces and there are two, so
total # of event when two dice are tossed: 62
5
Then P(sum is 8) =
and
36
5
P(sum is not 8) =1 
36
We do: (ex) On a certain day the chance of rain is 80% in San Francisco
and 30%in Sydney. Assume that the chance of rain in the two cities is
independent. What is the probability that it will not rain in either city?
A 7%
B 14%
(0.2)(0.7) = 0.14
C 24%
D 50%
We do: (ex) The probabilities that Jamie will try out for various sports
and team positions are shown in the chart below.
Jamie will definitely try out for either basketball or baseball, but not
both. The probability that Jamie will try out for baseball and try out for
catcher is 42%. What is the probability that Jamie will try out for
basketball? A 40%
B 60%
C 80%
D 90%
<try this> Two six-sided dice are rolled. Find the probability of the
given event.
a) The sum is greater than or equal to 5
b) The sum is less than or equal to 10.
c) The sum is greater than 2.
Probability of Independent & Dependent Events
Probability of Independent events
Definition:
An independent event is an event that is not affected by other event.
•If A and B are independent events, then the probability that both A and
B occur is:
P(A and B) = P(A)P(B).
I do (ex) You toss a die twice. What a probability to get 1 on the first
toss and 3 on the second toss.
1
1
P (to get 1) =
P (to get 3) =
6
6
1 1 1
P (to get 1 and 3) =     
 6   6  36
We do (ex) A basket has 5 red, 4 yellow, and 3 green. You are picking
3 balls with replacement. If you pick Red, Yellow, and Green in
order, you will be the winner. What is the probability to be a winner?
1st pick
5
P (red ) 
12
2nd pick
3rd pick
4
P( yellow) 
12
3
P( green) 
12
5  4  3 

Therefore, P( RYG )     
 12  12  12 
You do (ex) You randomly select two cards from a 52-card deck. What
is the probability that the first card is not a face and second card is a face
card with replacement.
<Try this> One bag contains 2 green marbles and 4 white marbles, and a
second bag contains 3 green marbles and 1 white marble. If you
randomly draw one marble from each bag, what is the probability that
they are both green?
Probability of Dependent events:
Definition:
A dependent event is an event in which one occurrence affects the other
occurrence. The probability that B will occur under that A has occurred
is called “Conditional probability” and noted by:
P(BA) read as “probability of B given A”
Formula:
If A and B are dependent events, then the probability that both A and B
occur is:
P(A and B) = P(A) P(BA)
I do (ex) (from CST sample question 2010)
A box contains 7 large red marbles, 5 large yellow marbles, 3 small
red marbles, and 5 small yellow marbles. If a marble is drawn at
random, what is the probability that it is yellow, given that it is one of
the large marbles?
P( yellow| large) =
# of large of yellow
Total # of large
We do (ex) You and two friends are at a restaurant for lunch. There are 8
dishes with the same price. What is the probability that each of you
orders a different dish?
Let three dishes are A, B, and C
1st person’s
P ( A) 
8
8
2nd person’s
7
P( B) 
8
 8  7  6 
P( A, B, &C )     
 8  8  8 
3rd person’s
6
P (C ) 
8
You do (ex) A math teacher is randomly distributing 15 plastic rulers
and 10 wooden rulers. What is the probability that the first ruler she
hands out will be a wooden one, and the second ruler will be a plastic
one?
1st pick
10
P ( wooden) 
25
2nd pick
15
P ( plastic) 
24
10  15 
P ( wooden & plastic )   
25  24 
(13 – 3 continued)
Probability Involving AND and OR
I do (ex)
Find the probability to draw a diamond or a face card from a deck
of cards.
13 diamond cards : P(D)
12 face cards: P(F)
3 diamond and face cards : P(D & F)
13 12 3
 
P(D or F) = P(D) + P(F) – P(D & F) =
52 52 52
We do: One card is drawn from a deck of cards. What is the
probability that the card is a black or an ace?
P(B)
P(A)
P(B & A)
P(B or A)
You do (ex)
One card is drawn from a deck of cards. What is the probability that
the card is either a face or heart?
P(F)
P(H)
P(F & H)
P(F or H)
Statistics:
Definition: Statistics are numerical values used to summarize and
compare sets of data. Measures of central tendency are:
1.Mean ( x
sum of the data values
Mean

):
number of data values
2.Median: Median  middle value or mean of the two middle value
x
3.Standard deviation: ia a measure of how each value in a data set
varies from the mean. And the notation is  (sigma)
the sume of (eadch data - mean value)2
=
n
n
or  =
2
(
X

X
)
 i
i 1
n
Where xi is each data as X 1 ,
, Xn
I do (ex) Find the mean and the standard deviation for the values:
48.0, 53.2, 52.3, 46.6, 49.9
48.0  53.2  52.3  46.6  49.9
X
 50.0
5
Mean
Standard deviation:
n
=
2
(
X

X
)
 i
i 1
n
(50  48.0)2  (50  53.2)2  (50  52.3)2  (50  46.6)2  (50  49.9)2

5
31.1

 2.5
5
We do (ex) Keith found the mean and standard deviation of the set of the
numbers given below. If he adds 5 to each number, what is the standard
deviation?
3, 6, 2, 1, 7, 5
For the data given:
24
X
4
6
28

6
After 5 is added:
54
X
9
6
(9  8) 2  (9  11) 2  (9  7) 2  (9  6) 2  (9  12) 2  (9  10) 2

6

28
6
<try this> Find the mean and the standard deviation for each set of
values.
(a)5, 6, 7, 3, 4, 5, 6, 7, 8
(b) 13, 15, 17, 18, 12, 21, 10