This document discusses principles of counting including factorial notation, the fundamental counting principle, permutations, and combinations. Factorial notation (n!) represents the product of all positive integers less than or equal to n. The fundamental counting principle states that if an activity can be completed in n1 ways and another in n2 ways, they can be completed together in n1*n2 ways. Permutations refer to arrangements of objects that consider order, while combinations only consider groupings without order. Formulas to calculate permutations and combinations are provided along with examples of problems applying these principles.

1. Principles of Counting
Factorial Notation
The Fundamental Counting Principle
The Counting Principle for
Alternative Cases
Permutation
Combination

2. Factorial Notation
For any positive integer n, n! means:
n (n – 1) (n – 2) . . . (3) (2) (1)
0! will be defined as equal to one
Examples:
4! = 4•3 •2 •1 = 24
The factorial symbol only affects the number it follows unless
grouping symbols are used.
3 •5! = 3 •5 •4 •3 •2 •1 = 360
( 3 •5 )! = 15! = big number

3. The Fundamental Counting
Principle
If activity 1 can be done in n1 ways, activity
2 can be done in n2 ways, activity 3 can be
done in n3 ways, and so forth; then the
number of ways of doing these activities on
a specified order is the product of n1, n2, n3
and so forth. In symbols,
nnnnn  321

4. Example 1:
Suppose a school has three
gates, in how many ways can
a student enter and leave
the school?

5. Example 2:
In a medical study, patients are
classified according to whether they
have blood type A, B, AB or O, and also
according to whether their blood
pressure is low, normal, or high. In
how many different ways can a patient
thus be classified.

6. Example 3:
A new car dealer offers a car in
four body styles, in ten colors, and
with a choice of three engines. In
how many ways can a person order
one of the cars?

7. Example 4:
A test consists of 15 multiple
choice questions, with each
question having four possible
answers. In how many different
ways can a student check off one
answer to each question?

8. Example 5:
How many different 4-digit
numbers can be formed from the
digits 1, 2, 3, 4 and 5 if: (a)
repetition is not allowed? How
many of these numbers are even?
How many are these numbers are
odd? (b) repetition is allowed?

9. The Counting Principle of
Alternative Cases
Suppose the ways of doing an activity can be
broken down into several alternative cases where
each case does not have anything in common with
the other cases. If case 1 can be done in n1 ways,
case 2 can be done in n2 ways, case 3 can be done
in n3 ways, and so on, then the number of ways
the activity can be done is the sum of n1, n2, n3
and so on cases. In symbols,
nnnnn  321

10. Permutation
The term permutation refers to the
arrangement of objects with reference
to order or it may be defined as an
arrangement of all or part of a set of
objects.

11. Linear Permutation
The number of permutations of n
distinct objects taken all together is
n!.
Example:
How many different signals can be
made using five flags if all flags
must be used in each signal?

12. Permutation of n Elements Taken
r at a Time
The arrangement of n objects in a specific order using r
objects at a time is given by
where r < n
Example:
Suppose there are eight machines, but only three
spaces in the display room available for the
machines. In how many different ways can the 8
machines be arranged in the three available spaces
 !
!
rn
n
Prn



13. Circular Permutation
The arrangement of n objects in a circular
pattern is given by the formula
Example:
In how many ways can six persons be seated
around a circular table?
 !1 nP

14. Permutation of Things Not All
Different
The number of distinct permutations of n objects
of which r1 are alike, r2 are alike, r3 are alike, …
etc. is
Example:
How many different permutations can be made
from the letters of the word “STATISTICS”?
!!!!
!
321 nrrrr
n
P



15. Combination
Suppose we are interested only in the number of
different ways that r objects can be selected
from a given number of objects. If the order of
the objects is not important, the total number of
orders or arrangement is called combination. The
number of combinations of n objects taken r at a
time is denoted by nCr and is given by the
formula:
  !!
!
rrn
n
Crn



16. Example 1:
In order to survey the opinions of
costumers at local malls, a
researcher decides to select 5
malls from a certain area with a
total of 9 malls. How many
different ways can the selection
be made?

17. Example 2:
The general manager of a fast-
food restaurant chain must select
6 restaurants from 10 for a
promotional program. How many
different possible ways can this
selection be done?

18. Problem 1:
In how many ways can 5 people
line up for a group picture if (a)
two want to stand next to each
other? (b) two refuse to stand
next to each other?

19. Problem 2:
In how many ways can 8
beads be put together to
form a round bracelet?

20. Problem 3:
A committee of 5 people must
be selected from 5 accountants
and 8 educators. How many
ways can the selection be done
if there are 3 educators in the
committee?

21. Problem 4:
In a club there are 8 women
and 5 men. A committee of 4
women and 2 men is to be
chosen. How many possibilities
are there?

22. Problem 5:
A committee of 5 people must be
selected from 5 accountants and 8
educators. How many ways can the
selection be done if there are at
least 3 educators in the
committee?

23. Problem 6:
How many different triangles
can be formed using the
vertices of an octagon?
Pentagon? Hexagon?