Counting Techniques
Counting Techniques
• Multiplication Principle (also called the
Fundamental Counting Principle)
• Combinations
• Permutations
• Number of subsets of a given set
Multiplication Principle
Suppose there are n different decisions and
each decision has ri choices where i is
some number from 1 to n.
Then the overall total number of ways in
which those n decisions can be made is
the product r1  r2  r3    rn
Examples
• Suppose you have 7 different shirts, 5 different pairs of
pants and 3 pairs of shoes. How many outfits are
possible?
• Suppose there are 10 questions on a multiple-choice
exam and each question can be answered in 5 different
ways (A, B, C, D or E). How many ways are there to
complete the exam assuming every question is
answered?
Examples
• Suppose you have 7 different shirts, 5 different pairs of
pants and 3 pairs of shoes. How many outfits are
possible?
– Answer: 7*5*3=105 possible outfits
• Suppose there are 10 questions on a multiple-choice
exam and each question can be answered in 5 different
ways (A, B, C, D or E). How many ways are there to
complete the exam assuming every question is
answered?
– Answer: 510  9,765,625 ways to complete the exam.
Combinations
• The number of ways of choosing r distinct
objects from n distinct objects is given by
the formula
n!
nCr 
(n  r )! r!
• Note n! n  (n  1)  (n  2)    3  2 1
and 0! = 1
Examples
• How many ways can 3 movies be chosen
from a list of 5 movies?
• A committee consists of 10 people. How
many ways are there to form a coalition of
5 people from the committee?
Examples
• How many ways can 3 movies be chosen
from a list of 5 movies?
– Answer: 5 C3  5  4  3  10
3 2
• A committee consists of 10 people. How
many ways are there to form a coalition of
5 people from the committee?
– Answer:
10
C5  252
Permutations
• The number of ways of selecting r distinct
objects from n distinct objects and
rearranging those r objects is given by the
formula
n!
n Pr 
(n  r )!
Examples
• Suppose there are 10 movies playing in
the theater. How many ways are there of
selecting and ranking your favorite 3?
• There are 5 people in a coalition of voters.
How many ways are there to rearrange
those 5 people in distinct orderings?
Examples
• Suppose there are 10 movies playing in
the theater. How many ways are there of
selecting and ranking your favorite 3?
– Answer:
P  10  9  8  720
10 3
• There are 5 people in a coalition of voters.
How many ways are there to rearrange
those 5 people in distinct orderings?
– Answer: 5! = 120 ways
Number of Subsets
• Given a set with n elements, the number
of subsets of the given set is 2 n .
• Examples:
– Let A = {x, y, z}. How many subsets does A
have?
– Suppose a committee consists of 3 people.
How many possible coalitions can be formed
from this committee?
Number of Subsets
• Examples:
– Let A = {x, y, z}. How many subsets does A
have?
• Answer:
23  8
subsets
– Suppose a committee consists of 3 people.
How many possible coalitions can be formed
from this committee?
• Answer:
2 8
3
coalitions