Chapter 9, Discrete Mathematics
Discrete mathematics: examples include a point and counting events
Continuous mathematics: examples include a line segment, intervals of real numbers,
calculus concepts like limits
Section 9-1, Basic Combinatorics
For homework, show set-up using formulas and then use calculator
HW: page 649-651: Day 1: 1, 3, 5, 9, 11, 13, 15, 24, 25
Day 2: 17, 19, 22, 23, 27, 29, 30, 32, 34, 36 (use calc formula), 39, 42
I. Multiplication Principle of Counting
Example A. Pearson Precal Textbook page 643
A car manufacturing company asked 1000 people to view their latest two-door coupe available in
four colors (Alabaster White, Black, Cherry Red, and Dusk Blue). The viewers were asked to
list their color preferences in order from favorite to least favorite.
1. How many ways could the [individual lists of] colors be ordered?
2. Of the 1000 viewers, how many would you expect to choose the preference order
Alabaster White, Black, Cherry Red, and Dusk Blue?
Let’s analyze the tree diagram on the next page to answer these two questions.
Alabaster White
Black
Cherry Red
Dusk Blue
Black
Alabaster White
Cherry Red
Dusk Blue
Cherry Red
Black
Alabaster White
Dusk Blue
Dusk Blue
Black
Alabaster White
Cherry Red
Cherry Red
Dusk Blue
Dusk Blue
Cherry Red
Black
Dusk Blue
Dust Blue
Black
Cherry Red
Black
Black
Cherry Red
Cherry Red
Dusk Blue
Dusk Blue
Cherry Red
Alabaster White
Dusk Blue
Dust Blue
Alabaster White
Cherry Red
Alabaster White
Alabaster White
Cherry Red
Alabaster White
Dusk Blue
Dusk Blue
Alabaster White
Black
Dusk Blue
Dust Blue
Black
Alabaster White
Black
Black
Alabaster White
Alabaster White
Cherry Red
Cherry Red
Alabaster White
Black
Cherry Red
Cherry Red
Black
Alabaster White
Black
Black
Alabaster White
Answers:
1. Total: _______________ lists of different color rankings
2.
Do you want to do a tree diagram to figure out the possible order of 10 colors?
Why or why not?
The tree diagram is a geometric visualization of a fundamental counting principle:
Multiplication Principle of Counting
If a procedure P has a sequence of stages, S1,
S2, …, Sn and if
S1 can occur in r1 ways,
S2 can occur in r2 ways,
.
.
.
Sn can occur in rn ways,
Then the number of ways that the procedure P can occur is the
product r1 r2 …rn .
Example B: Apply the Multiplication Principle of Counting and analyze the exterior and interior
colors of the 2012 VW Passat to determine the number of different exterior and interior color
combinations.
Source: http://www.cars.com/volkswagen/passat/2013/colors
2013 Volkswagen Passat
Exterior Colors
Black
Candy White
Glacial Blue Metallic
Night Blue Metallic
Opera Red Metallic
Platinum Gray Metallic
Reflex Silver Metallic
Tungsten Silver Metallic
Interior Colors
Beige
Moonrock
Titan Black
II. Permutations (synonym: orderings)
A. Applying the Multiplication Principle of Counting gives the number of ways that a set of n
objects (an n-set) can be arranged in order. Each such ordering is called a permutation of the set.
There are n! permutations of an n-set
If n is a positive integer n, then the symbol n! (read “n factorial”) represents the product:
n(n – 1)(n – 2)(n – 3)…2 1
Note: 0! = 1
and
1! = 1
Applying the Multiplication Principle of Counting to the ranking list of the four car
colors in the preceding Example A shows:
Student Example: In how many orders can three girls and two boys walk through a classroom
doorway single file if there are not restrictions on which sex goes first?
B. Permutation of n elements taken r at a time
Example: A typical field of twenty horses runs in the Kentucky Derby at Churchill Downs.
How many different ways can the colts come in first, second, and third place? Assume that there
are no ties.
Applying the Multiplication Principle of Counting:
19 colts ran the 2013 Kentucky Derby (one colt scratched, and two other sub colts were not
ready to jump into the field):
http://www.youtube.com/watch?v=70BAs0p9enA
In example B, we are ordering a subset of a collection of elements (here, the elements are
racehorses) rather than the entire collection (the field of 19 colts). This kind of ordering is called
a permutation of n elements taken r at a time.
We can use a handy formula for the permutation of n elements taken r at a time:
Please Memorize!
n
Pr =(
0≤r≤n
for
)
Ordered
Applying this formula to the Kentucky Derby problem in example B:
___
P __ =
C. Distinguishable (synonym: different) Permutations
Suppose we want to find the possible permutations of the letters A, A, B, and C.
Using the formula: 4
P4 =
(
=
)
( )
=
=
But some permutations would not be distinguishable because there are two A’s in the list.
Distinguishable Permutations
There are n! distinguishable permutations of an n-set containing n distinguishable objects.
If an n-set contains n1 objects of a first kind,
n2 objects of a second kind, and so on, with
n1 + n2 + … nk = n, then
The formula for the number of distinguishable permutations of the n-set is:
Memorize; or just know how to use!
The *distinguishable permutations of the letters A, A, B, and C =
=
=
= 12
*The two letter A’s give us permutations that are the same so we want to reduce the number of
available choice by dividing by 2!
A list of the distinguishable permutations of the letters A, A, B, and C confirms this result, but
this list took me quite some time to figure out by writing the orderings:
AABC
BAAC
AACB
BACA
ABCA
BCAA
ACBA
CAAB
ABAC
CABA
ACAB
CBAA
Student Examples
1: Use your calculator to count the number of different 11-letter words (don’t worry about
whether they’re in the dictionary) that can be formed by using the letters in CHATTANOOGA.
2. Evaluate without a calculator, showing all work:
a) 6
P4
b) 5
Pn
for n ≤ 5
c) n
P5
for 0 ≤ 5 ≤ n
III. Combinations
Combinations are subsets of a larger set in which order is not important. These unordered
subsets are called combinations of n objects taken r at a time.
Example A
Suppose we randomly select a subset of three pool balls from a set of 16 pool balls, and we
happen to choose balls 1, 2 and 3.
These are the possibilities:
6 Permutations: Order does matter
1 Combination: Order doesn’t matter
123
132
213
231
312
321
123
Combination Counting Formula:
The formula for the number of combinations of n objects taken r at a time is:
Please Memorize!
n
Cr =
(
)
for
0≤r≤n
Unordered
Alternative notation read as “n choose r:”
Example B: A standard poker hand consists of 5 cards dealt from a deck of 52 cards. How
many different poker hands are possible? (After the cards are dealt, the players may reorder their
hand, and therefore order is not important.)
Students, Example C: In order to conduct an experiment, four students are randomly selected
from a class of 20. How many different groups of four students are possible?
IV. Subsets of an n-set
The formula for Counting Subsets of an n-set is:
Memorize; or just know how to use!
n
There are 2 subsets of a set with n objects (including the empty set and the entire set)
Example A, textbook on page 647: Armando’s Pizzeria offers patrons any combination of up to
10 different toppings: pepperoni, mushrooms, sausage, onions, green peppers, bacon, ham, black
olives, green olives, and anchovies.
How many different pizzas can be ordered
1) if we choose any 3 toppings?
2) if we can choose any number of toppings (0 through 10)?
Questions to ponder for 1):
Does order matter? Is a pizza with green peppers and mushrooms the same as a pizza
with mushrooms and green peppers?
Is this a permutation or combination?
What is the formula to be used?
Solution work for 1:
For number 2, let’s think about applying the formula for combinations 10
combinations.
r = 0 corresponds to zero toppings
r = 1 corresponds to one topping
r = 2 corresponds to two toppings
Cr
for all the
.
.
.
r = 10 corresponds to all 10 toppings
We could add all of these combinations together, but this would be tedious:
10
C 0 + 10 C 1 + 10 C 2 + … 10 C 10
An easier solution to this problem would be to apply the Multiplication Principle of Counting.
Let’s scroll back in our notes, and reread this property.
For each of the 10 toppings (stages S1, S2, …, S10) for the pizza’s various toppings (the finished
pizza is the Procedure), we have two ways these occur (yes or no).
pepperoni, mushrooms, sausage, onions, green peppers, bacon, ham, black olives, green olives,
and anchovies
A pizza with green peppers and mushrooms would correspond to the sequence:
NYNNYNNNNN
Per the Multiplication Principle of Counting:
which is the number of different pizzas that could be ordered.
= 210 = 2^10 = 1024,
Or we could use the formula for counting subsets of the set of 10 pizza toppings: 210 = 1024
Student Example B:
Calculate the number of possible local telephone numbers in America for any given area code.
Suppose that any number 0 through 9 could be used in any of the seven digit positions, including
zero.
_____ _____ _____ - _____ _____ _____ _____
Apply the Multiplication Principle of Counting to find the number of unique telephone numbers
that are theoretically possible.
Now, calculate the number of possible local telephone numbers in America for any given
area code excluding zero from the first digit.
Student Example C: How many different ways can a true-false test be answered if it has 20
questions?
A few parting thoughts…
Permutations
Combinations
Factorials
are for lists
order matters
position is important
are for groups
order doesn’t matter
fewer combinations than
permutations for n, r
means to multiply a
series of descending
natural (counting)
numbers
Graphing Calculator Keystrokes:
12
P3
16
Press:
C4
8!
Press:
Press:
12
16
8
Math ► ► ► PRB
Math ► ► ► PRB
Math ► ► ► PRB
2: nPr
3: nCr
4: !
3
4
ENTER
Answer: 1320
Answer: 1820
Answer: 40,320