2. Chapter 11: Further Topics in Algebra
11.1 Sequences and Series
11.2 Arithmetic Sequences and Series
11.3 Geometric Sequences and Series
11.4 Counting Theory
11.5 The Binomial Theorem
11.6 Mathematical Induction
11.7 Probability

3. 11.4 Counting Theory Example If there are 3 roads from Albany to Baker
and 2 Roads from Baker to Creswich, how many
ways are there to travel from Albany to Creswich?
Solution The tree diagram shows all possible
routes.
For each of the 3 roads to Albany, there are 2 roads to Creswich. There are 3.2=6 different routes from Albany to Creswich.

4. 11.4 Counting Theory
Two events are independent if neither influences the outcome of the other. For example, choice of road to Baker does not influence the choice of road to Creswich.
Sequences of independent events can be counted using the Fundamental Principle of Counting.

5. 11.4 Counting Theory Fundamental Principle of Counting
If n independent events occur, with
m1 ways for event 1 to occur,
m2 ways for event 2 to occur, …
and mn ways for event n to occur,
then there are
different ways for all n events to occur.

6. 11.4 Using the Fundamental Principal of Counting
Example A restaurant offers a choice of 3
salads, 5 main dishes, and 2 desserts. Count the
number of 3-course meals that can be selected.
Solution The first event can occur in 3 ways,
the second event can occur in 5 ways, and the
third event in 2 ways; thus there are possible meals.

7. 11.4 n-Factorial n-Factorial For any positive integer n, By definition
Example Evaluate (a) 5! (b) 7!
Solution (a)
(b)

8. 11.4 Permutations
A permutation of n elements taken r at a time is one of the arrangements of r elements from a set of n elements.
The number of permutations of n elements taken r at a time is denoted P(n,r).
Example An ordering of 3 books selected from 5
books is a permutation. There are 5 ways to select
the first book, 4 ways to select the second book, and
3 ways to select the third book. Thus, there are
permutations.

9. 11.4 Permutations Permutations of n Elements Taken r at a Time
If P(n,r) denotes the number of permutations of n elements taken r at a time, with r < n, then

10. 11.4 Using the Permutation Formula
Example Suppose 8 people enter an event in a
swim meet. In how many ways could the gold,
silver, and bronze prizes be awarded.
Solution
ways

11. 11.4 Combinations
A subset of items selected without regard to order is called a combination.
The number of combinations of n elements taken
r at a time is denoted C(n,r) or nCr and is sometimes read as “n choose r”.

12. 11.4 Combinations Combinations of n Elements Taken r at a Time
If C(n,r) represents the number of combinations of n things taken r at a time, with r < n, then

13. 11.4 Using the Combinations Formula
Example How many different committees of 3
people can be chosen from a group of 8 people?
Solution Since a committee is an unordered set
(the arrangement of the 3 people does not
matter,) the number of committees is

14. 11.4 Distinguishing between Permutations and Combinations
Different orderings or arrangements of the r objects are different permutations.
Clue words: arrangement, schedule, order
Each choice or subset of r objects gives one combi-nation. Order within the group of r objects does not matter.
Clue words: group, committee, sample, selection