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