Ch. 11.6
Counting Principles
Example 1
 Eight pieces of paper are numbered from
1-8 and placed in a box. One piece of
paper is drawn from the box, its number
is written down, and the piece of paper is
replaced in the box. Then, a second
piece of paper is drawn from the box and
its number is written down. Finally, the
two numbers are added together. How
many different ways can a sum of twelve
be obtained????
Example 2
 Eight pieces of paper are numbered 1-8
and placed in a box. Two pieces of paper
are drawn at random from the box at the
same time, and the numbers on the
pieces of paper are written down and
totaled. How many different ways can a
sum of 12 be obtained.
Ex 1 and 2
 The difference between examples 1 and
2 is that example 1 occurred WITH
REPLACEMENT and example 2
occurred WITHOUT REPLACEMENT
(this eliminates extra possibilities)
Fundamental Counting
Principle
 Let E1 and E2 be events. The first event
E1 can occur in m1 different ways. After
E1 has occurred, E2 can occur in m2
different ways. The number of ways that
the two events can occur is m1 m2
Example 3
 How many different pairs of letters from
the English alphabet are possible?
Example 4
 Telephone numbers in the United States
currently have 10 digits. The first three
are the area code and the next seven are
the local telephone number. How many
different telephone numbers are possible
within each are code? (Note that at this
time, a local telephone number cannot
begin with 0 or 1)
Permutations
 Used to determine “the number of ways
that n elements can be arranged”
 Def: permutation – a permutation on n
elements is an ordering of the elements
such that one element is first, on is
second, on is third and so on
Example 5
 How many permutations are possible for
the letters A, B, C, D, E, and F??
Example 6
 Eight horses are running in a race. In
how many different ways can these
horses come in first, second or third?
Permutations of n Elements
Taken R at a time
 The number of permutations of n
elements taken r at a time is
P(n,r) = __n!__
(n-r)!
Distinguishable
Permutations
 Suppose a set of n objects has n1 of one
kind of object, n2 of a second kind and so
on with
n = n1+n2+n3+n4 +….nk
Then the number of distinguishable
permutations of the n objects is
______n!_____
n1!n2!n3!....nk!
Example 7
 In how many distinguishable ways can
the letters in BANANA be written?
Combinations
 Selecting items of a larger set in which
order is NOT important
The number of combinations of n
elements taken r at a time is
C(n,r) = __n!__
(n-r)! r!
Example 8
 In how many different ways can three
letters be chosen from the letters A, B, C,
D, and E?
Example 9
 A standard poker hand consists of five
cards dealt from a deck of 52 cards.
How many different poker hands are
possible?
(After the cards are dealt, the player may
reorder them, so order is not important)
Example 10
 You are forming a 12 member swim team
from 10 girls and 15 boys. The team
must consist of five girls and seven boys.
How many different 12 member teams
are possible?
Homework
 Pg 860 #7-17, 21,37-39, 41,42, 49-55