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:

The number of permutations of a set with n elements is P(n, n) = n!

(a, b, c) (b, a, c) (c, a, b)

(a, c, b) (b, c, a) (c, b, a)

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)?