8.6 Counting Principles
Listing Possibilities: Ex 1
• Eight pieces of paper are numbered from 1 to 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 piece of paper
is again drawn from the box, and its number written
down. Finally, the two numbers are added together.
How many different ways can a total of 12 be
obtained?
Listing Possibilities: Ex 2
• Eight pieces of paper are numbered from 1 to 8 and
placed in a box. Two pieces of paper are drawn from
the box, and the numbers on the paper are written
down and totaled. How many different ways can a
total of 12 be obtained?
The Fundamental Counting Principle
• Let E1 and E2 be two 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
ways.
The Fundamental Counting Principle Ex 3
• How many different pairs of letters from the English
alphabet are possible?
The Fundamental Counting Principle Ex 4
• How many different telephone numbers are possible
within each area code? (Note: local telephone
numbers cannot begin with a 0 or 1)
Permutations
(When Order Matters)
• A permutation of n different elements such
that one element is first, one is second, one is
third, and so on.
Ex 5: How many permutations are possible for the letters A, B, C,
D, E, and F?
Permutations
• The number of permutations of n
elements is given by
Ex 6: Eight horses are running a race. In how many ways can
these horses come in first, second, and third? (Assume
there are no ties)
Permutations
• The number of permutations of n elements taken
r at a time is
• Distinguishable Permutations:
• Suppose a set of n objects has n1 of one kind
of object, n2 of a second kind, n3 of a third
kind, and so on, with n=n1+n2+…+nk . Then
the number of distinguishable permutations is
Permutations: Ex 7
• In how many distinguishable ways can the letters in
BANANA be written?
Combinations
When Order Is Not Important
• Ex 8: In how many different ways can three letters
be chosen from the letters A, B, C, D, and E? (The
order of the three letters is not important.)
Combinations
• The number of combinations of n elements
taken r at a time is
Ex 9: A standard poker hand consists of five cards dealt from a
deck of 52. How many different poker hands are possible?