The Generalized Product Rule
Counting Permutations
ICS 6D
Sandy Irani
• Student council has 15 members. Must select
officers (Pres, VP, Treasurer, Secretary)
• No one can serve in more than one officer
position.
• How many ways to select offiers?
( ______ , ______ , ______ , ______ )
[President]
[Vice President]
[Treasurer]
[Secretary]
Counting as a Selection Process
• Select an element from a set to be counted
• Selection process is a sequence of steps. In each
step, one more decision is made about the item
to be selected.
• At the end of the process, the item to be selected
is fully specified.
• If at each step, the number of choices is
independent of the previous choices made
– The number of items in the set is equal to the product
of the number of choices at each step.
Generalized Product Rule
• Set S of sequences of k items.
• Suppose there are:
– n1 choices for the first item.
– For every possible choice for the first item, there are
n2 choices for the second item.
– For every possible choice for the first and second
items, there are n3 choices for the third item.
– For every possible choice for the first k-1 items, there
are nk choices of the kth item.
• Then |S|=n1· n2 · · · · nk
Generalized Product Rule
• A group of software engineers must complete three modules
in a software project. One programmer must be assigned to
each module and there are 10 programmers in the group.
• How many ways are there to select the programmers to write
the different modules?
( _________ , _________ , _________ )
[Module 1]
[Module 2]
[Module 3]
What if each programmer can be assigned to at most one module?
Generalized Product Rule
• A group of software engineers must complete three modules in a software
project. One programmer must be assigned to each module and there are
10 programmers in the group.
• 7 of the members of the group are senior employees and 3 are
junior employees.
• Module 1 must be written by a senior employee and module 2 must
be written by a junior employee.
• How many ways are there to select the programmers to write the
different modules?
( _________ , _________ , _________ )
[Module 1]
[Module 2]
[Module 3]
What if each programmer can be assigned to at most one module?
r-Permutations
• Definition: an r-permutation is a sequence of r
items with no repetitions selected from the same
set.
– Example: S = {a, b, c, d, e, f, g, h, i}
– (g, a, b, e) is a 4-permutation
– (a, e, b, g) is a different 4-permutation (order matters)
• How many ways are there to select an
r-permutation from a set of n elements?
Counting r-permutations
• The number of ways to select an
r-permutation from a set of n items is:
n·(n-1) ·(n-2) ···(n-r+1)
Counting r-permutations
• The number of ways to select an
r-permutation from a set of n items is:
n·(n-1) ·(n-2) ···(n-r+1) =
𝑛!
(𝑛−𝑟)!
= P(n,r)
Counting r-permutations
• Four different tasks are distributed to computers in a
distributed system of 20 computers.
– How many ways are there to assign the tasks to computers
with no restrictions on the number of tasks assigned to any
one computer?
– How many ways are there it assign the tasks if each
computer gets assigned at most one task?
Counting r-permutations
• Three kids (Larry, Curly, and Moe) select prizes
from a bin with 30 different prizes. How many
ways are there for the kids to select the prizes?
Permutations
• A permutation is a sequence that contains
each element of a finite set exactly once
• Example: S = {a, b, c}. The permutations of S
are:
(a, b, c)
(b, a, c)
(c, a, b)
(a, c, b)
(b, c, a)
(c, b, a)
The number of permutations of a set with n
elements is P(n, n) = n!
Counting Permutations
• The number of permutations of the set
S = {a, b, c, d, e} is:
How many permutations in which b and e are
next to each other?
Counting Permutations
• The number of permutations of the set
S = {a, b, c, d, e} is:
How many permutations in which b comes
immediately before e?
Counting Permutations
• The number of permutations of the set
S = {a, b, c, d, e} is:
How many permutations in which b comes
somewhere before e (but not necessarily
immediately before e)?