Counting Techniques • Tree Diagram • Multiplication Rule • Permutations • Combinations
Counting Techniques
• Tree Diagram
• Multiplication Rule
• Permutations
• Combinations
Tree Diagram
a method of listing outcomes of an experiment consisting of a series
of activities
Tree diagram for the experiment of tossing two coins
start
H
T
H
H
T
T
Find the number of paths without constructing the tree
diagram:
Experiment of rolling two dice, one after the other and observing any of the six possible outcomes each time .
Number of paths = 6 x 6 = 36
Multiplication of Choices
If there are n possible outcomes for event E1
and m possible outcomes for event E2,
then there are n x m or nm possible outcomes
for the series of events E1 followed by E2.
Area Code Example
Until a few years ago a three-digit area code was designed as follows.
The first could be any digit from 2 through 9.The second digit could be only a 0 or 1.
The last could be any digit.How many different such area codes were
possible? 8 2 10 = 160
Ordered Arrangements
In how many different ways could four items be arranged in order from first to last?
4 3 2 1 = 24
Factorial Notation
• n! is read "n factorial"
• n! is applied only when n is a whole
number.
• n! is a product of n with each positive
counting number less than n
Calculating Factorials
5! = 5 • 4 • 3 • 2 • 1 =
3! = 3 • 2 • 1 =
120
6
Definitions
1! = 1
0! = 1
Complete the Factorials:
4! =
10! =
6! =
15! =
24
3,628,800
720
1.3077 x 1012
Permutations
A permutation is an arrangement in a particular order of a group of items.
There are to be no repetitions of items within a permutation.)
Listing Permutations
How many different permutations of the letters a, b, c are possible?
Solution: There are six different permutations:
abc, acb, bac, bca, cab, cba.
Listing Permutations
How many different two-letter permutations of the letters a, b, c, d are possible?
Solution: There are twelve different permutations:
ab, ac, ad, ba, ca, da, bc, bd, cb, db, cd, dc.
Permutation Formula
The number of ways to arrange in order n distinct objects, taking them r at a time, is:
!rn!nP r,n
Another notation for permutations:
rn P
Find P7, 3
21024
5040!4!7
)!37(!7P 3,7
Applying the Permutation Formula
P3, 3 = _______ P4, 2 = _______
P6, 2 = __________ P8, 3 = _______
P15, 2 = _______
6 12
30 336
210
Application of Permutations
A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen and arranged in order from #1 to #5?
Solution: P8,5 = !3!8
= 8• 7 • 6 • 5 • 4 = 6720
Combinations
A combination is a grouping in no particular order
of items.
Combination Formula
!r!)rn(!nC r,n
The number of combinations of n objects taken r at a time is:
Other notations for combinations:
rn
orCrn
Find C9, 3
84)720(6
362880!6!3
!9)!39(!3
!9C 3,9
Applying the Combination Formula
C5, 3 = ______ C7, 3 = ________
C3, 3 = ______ C15, 2 = ________
C6, 2 = ______
35
1 105
10
15
Application of Combinations
A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen if order makes no difference?
Solution: C8,5 = !3!5!8
= 56