Permutations and Combinations Rosen, Chapter 5.3 Motivating - - PDF document
10/17/2016 Permutations and Combinations Rosen, Chapter 5.3 Motivating question In a family of 3, how many ways can we arrange the members of the family in a line for a photograph? 1 10/17/2016 Permutations A permutation of a set of
In a family of 3, how many ways can we
A permutation of a set of distinct objects is an
Example: (1, 3, 2, 4) is a permutation of the
How many permutations of n objects are
How many permutations of n objects are
Using the product rule:
Anagram: a word, phrase, or name formed by
The anagram server: http://wordsmith.org/anagram/
How many ways can we arrange 4 students
How many ways can we arrange 4 students
How many ways can we select 3 students
How many ways can we select 3 students
permutation – a permutation of a set of
r-permutation – a ordered arrangement of r
You invite 4 people for a dinner party. How
A) 6 (3!) B) 24 (4!) C) 120 (5!) D) 16 (n2) E) 32 (2n2)
You invite 4 people for a dinner party. How
A) 6 (3!) B) 24 (4!) C) 120 (5!) D) 16 (n2) E) 32 (2n2)
In how many ways can a photographer at a
In how many ways can a photographer at a
6! = 720
In how many ways can a photographer at a
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B.
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In how many ways can a photographer at a
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B.
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In how many ways can a photographer at a
In how many ways can a photographer at a
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B.
C.
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In how many ways can a photographer at a
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B.
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In how many ways can a photographer at a
6! possible ways for 6 2*5! - possible ways the bride is next to the
In how many ways can a photographer at a
In how many ways can a photographer at a
5! + (4 · 4!) + (3 · 4!) + (2 · 4!) + (1 · 4!) =
The first position to fill is the position of the
5! + (4 · 4!) + (3 · 4!) + (2 · 4!) + (1 · 4!) =
Count the number of ways to arrange n men
Count the number of ways to arrange n men
Count the number of ways to arrange n men
An optimal TSP tour through Germany’s 15 largest cities
Go through all permutations of cities, and
Need a method for generating all
Note: how many solutions to a TSP problem
Let's design a recursive algorithm for
Let's design a recursive algorithm for
Starting point: decide which element to put first what needs to be done next? what is the base case?
Is our algorithm for TSP that considers all
An ordered arrangement of r elements of a set:
Example: List the 2-permutations of {a,b,c}.
The number of r-permutations of a set of n elements: then
How many ways are there to select a first
How many ways are there to select a first
How many permutations of the letters
How many permutations of the letters
How many permutations of the letters
Suppose there are 8 runners in a race. The
Suppose there are 8 runners in a race. The
How many poker hands (five cards) can be
How is this different than r-permutations?
An r-combination of a set is a subset of size r The number of r-combinations out of a set
{1,3,4} is a 3-combination of {1,2,3,4} How many 2-combinations of {a,b,c,d}?
How many r-combinations?
C(n,r) satisfies:
We can see that easily without using the formula
Two ordered selections are the same if
the elements chosen are the same; the elements chosen are in the same order.
Ordered selections: r-permutations. Two unordered selections are the same if
the elements chosen are the same.
(regardless of the order in which the elements are chosen)
Unordered selections: r-combinations.
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Suppose we want to compute P(n,r) . Constructing an r-permutation from a set of n elements
Step 1 can be done in C(n,r) different ways. Step 2 can be done in r! different ways. Based on the multiplication rule, P(n,r) = C(n,r) ∙ r! Thus
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How many poker hands (five cards) can be
How many poker hands (five cards) can be
There are 52! / 47! Permutations so
Since order doesn’t matter, there are 5!
The faculty in biology and computer science want to
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The faculty in biology and computer science want to
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There are C(20,2) combinations of CS There are C(30,2) combinations of Biology Using the product rule the total combinations
A coin is flipped 10 times, producing
A coin is flipped 10 times, producing
A coin is flipped 10 times, producing
A coin is flipped 10 times, producing
A coin is flipped 10 times, producing
A coin is flipped 10 times, producing
A coin is flipped 10 times, producing
1 way for 0 tails 10 ways for 1 tail 10!/2!*8! = 90/2 = 45 ways for 2 tails 10!/3!*7! = 720/6 = 120 ways for 3 tails
A coin is flipped 10 times, producing
A coin is flipped 10 times, producing
While commanding the Enterprise, Spock has
How can we do this?
consider the nth object
C(k, k) = 1
C(n, 0) = 1
This is an example of a recurrence relation, which is a
Apply the multiplication rule if
The elements to be counted can be obtained through
Each step is performed in a fixed number of ways
Apply the addition rule if
The set of elements to be counted can be broken up
Apply the inclusion/exclusion rule if
It is simple to over-count and then to subtract
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Make sure that
When using the addition rule:
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