This document discusses counting techniques including permutations and combinations. It provides examples of using the fundamental counting principle to calculate the number of possible outcomes in situations like selecting shirts from various sizes and colors or generating license plates from letters and numbers. The key concepts of permutations, which involve ordered arrangements, and combinations, which involve unordered selections, are explained along with related formulas. Practice problems apply these techniques to scenarios such as seating arrangements, word rearrangements, and group selections.

1. Counting
Techniques,
Permutations
and
Combinations

2. Fundamental
Counting
Principle

3. The Fundamental Counting
Principle (FCP)


4. Example 1
The Shirt Mart sells shirts in sizes S, M, L, and
XL. Each size comes in five colors: red,
yellow, white, orange, and blue. The shirts
come in short sleeve and long sleeve. How
many kinds of shirts are there?

5. Example 2
A plate number is made up of two
consonants followed by three nonzero digits
followed by a vowel. How many plate
numbers are possible if
a. The letters and digits cannot be
repeated in the same plate number?
b. The letters and digits can be repeated in
the same plate number?

6. Example 3
Carla is taking a matching test in which he is
supposed to match four answers with four
questions. In how many different ways can
he answer the four questions?

7. Practice Exercises
1. How many ordered pairs of letters are there that use only the
letters A, B, C, D, and E?
2. How many different sequences of heads or tails are possible if
a coin is flipped 8 times?
3. A model is selecting her outfits purchased 5 blouses, 4 skirts and
3 blazers. How many different new outfits consisting of a
blouse, a skirt and a blazer can she create from her new
collection?
4. Five different mathematics books and six different grammar
books are to be arranged on a shelf. How many possible
arrangements can be made if
a. the books on the same subjects are to be arranged
together?
b. the books are to be arranged alternately?
5. From the word “ALERT,” determine how many letter
arrangements are possible given the following conditions:
a. All 5 letters are used without restrictions
b. all vowels and consonants are together
c. Only three letters are used without repeating any letter.
d. Only four letters are used without restrictions.

8. Permutation

9. Factorial Notation
In general, if n is a positive integer, then n
factorial denoted by n! is the product of all
integers less than or equal to n.
n! = n. (n-1).(n-2). … . 2.1
As a special case, we define
0! = 1

10. Definition
A permutation is the ordered arrangement
of distinguishable objects without allowing
repetitions among the objects.

11. 

12. Example 1
In how many ways can a president and
vice-president be chosen from a club with
12 members?

13. If there are 50 floats in Penagbenga
Festival, how many ways can a first-place, a
second-place, and a third-place trophy be
awarded?

14. Example 2
Find the number of different arrangments of
the set of six letters HONEST
a. Taken two at a time
b. Taken three at a time
c. Taken six at a time.

15. Example 3
Find the number of permutations in each
situation.
a. A softball coach chooses the first,
second, and third batters for a team of
10 players.
b. Three-digit numbers are formed from the
digits 2, 3, 4, and 5, with no digits
repeated.

16. Example 4
Five golfers on a team are playing in a
tournament. How many different line-up
can the coach make?

17. Example 5
Find the number of ways a president, a
vice-president, a secretary, and a treasurer
can be chosen from among Alvin, Aris,
Richard, Ricky, Alma, and Alice.

18. Permutations of Identical objects


19. Practice Exercises
In how many different ways can four
people be seated in a row?

20. Example 1
In how many ways may the letters of the
word “STATISTICS” be arranged?

21. Example 2
In how many ways may the letters of the
word “ASSESSMENT” be arranged?

22. Circular Permutations


23. Example
In how many ways may seven persons be
seated around a circular table?

24. Circular Permutations with no
definite top or bottom


25. Example
In how many ways can 3 keys be arranged
in a key ring?

26. More exercises?
Visit www.khanacademy.org , join the stat
class 2013 and add me as coach (code:
Q87N9A) for me to track your progress and
give additional points.

27. Quiz
1. In how many ways may the letters of the
word ASSESSMENT be arranged?
2. How many license plates can be
manufactured with three letters followed
by three digits? No repetition
3. How many distinguishable ways can 4
beads be arranged in a circular
bracelet?
4. In how many ways can 9 people be
seated in a round table?
5. How many permutations of letters in the
word GOOGOLPLEX?

28. Combinations

29. Combination


30. Example 1
Ellen received an offer to join a CD club. If
she agrees to be a member, she can select
5 CDs from a list of 40 CDs. In how many
ways can Ellen select the 5 CDs?

31. Example 2
Mr. Elton has to choose three of the six
officers of the Math Club to go to a regional
meeting. How many possible choices does
he have?

32. Example 3
How many subcommittees of 5 people can
be formed from a committee consisting of 8
people?

33. Example 4
A class consists of 12 boys and 15 girls. How
many different committees of four can be
selected from the class if each committee is
to consist of two boys and two girls?

34. Practice Exercise
1. How many combinations of 5 records
can be chosen from 12 records offered
by a record club?
2. How many choices of 5 pocketbooks to
read can be made from a set of nine
pocketbooks?
3. A math professor gave his class a
problem set consisting of 10 problems
and required each student to answer
any 7 problems. In how many ways can
a student choose 7 problems from the
problem set?