Summary of Different Permuations and Combinations
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Order matters?
Combinatorics and
Graph Theory
Question: How many ways to pick r objects from . . .
n objects
n types of objects
Permutation
Yes
P (n, r ) =
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C (n, r ) =
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Permutations with Indistinguishable Objects
How many different strings can be made by reordering the
letters in the word ALL?
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This is not given by 3! because some of the letters in this
word are the same
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Different strings: ALL, LAL, LLA ⇒ 3 possibilities, not 6
because relative ordering of L’s doesn’t matter
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In general, how can we compute the number of permutations
of n objects where some of them are indistinguishable?
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n!
r !·(n−r )!
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C ∗ (n, r ) =
(n+r −1)!
r !·(n−1)!
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Consider n objects such that:
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n1 of them indistingishable of type 1
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n2 of them indistingishable of type 2
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...
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nk of them indistinguishable of type k
The number of permutations in this case is given by:
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Proof, cont.
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Let’s decompose using product rule:
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First, place all n1 objects of type 1
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Then all n2 objects of type 2 etc.
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How many ways to place n1 indistinguishable objects in n
slots?
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Combination w/ repetition
n!
n1 !n2 ! . . . nk !
Proof
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P ∗ (n, r ) = n r
Permutations with Indistinguishable Objects, cont.
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n!
(n−r )!
Combination
No
Permutation w/ repetition
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1
Now, how many ways to place n2 objects of type 2?
Continuing this way and using product rule, number of
permutations is:


kP
−1
n
n − n1
n − n1 − n2
n−
ni 

·
·
...
i=1
n2
n3
n1
nk
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Proof, cont.
Example 1
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n
n1


n − n1
n−
ni 
·
...
i=1
n2
nk
kP
−1
Let’s expand this definition:
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How many different strings can be made by ordering the
letters of the word SUCCESS?
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n!
n1 !n2 ! . . . nk !
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This simplifies to:
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Another way to see this: Compute total # of permutations (n!) and
then divide by # of relative orderings between objects of type 1
(n1 !), # of relative orderings of objects of type 2 (n2 !) etc.
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Example 2
There are 3 identical red balls, 5 identical blue balls, and 2
identical green balls.
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In how many different ways can these balls be arranged if the
first ball must be blue?
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Many counting problems can be thought of as distributing
objects into boxes
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In some cases both objects and boxes are distinguishable
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In some cases, boxes are distinguishable, but objects are
indistinguishable
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Example: Distinguishable Objects in Distinguishable Boxes
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Distributing Objects into Boxes
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Example, cont.
How many ways are there to distribute 5 cards to each of 4
players from a deck of 52 cards?
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Indistinguishable Objects Into Distinguishable Boxes
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Distributing Objects into Boxes Summary
How many ways are there to place 10 indistinguishable balls
into 8 distinguishable bins?
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Distinguishable objects into distinguishable boxes:
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Involves permutations
Indistinguishable objects into distinguishable boxes:
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Involves combination
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Another Example
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Example 2
How many ways to distribute six distinguishable objects to five
distinguishable boxes?
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How many ways to distribute six indistinguishable objects to
five distinguishable boxes?
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Example 3
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New Topic: Graphs
How many ways to assign 15 distinguishable objects into 5
distinguishable boxes so boxes contain 1, 2, 3, 4, 5 objects
respectively?
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Graph is a fundamental mathematical structure
in computer science
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Graph G = (V , E ) consists of a set of vertices
(nodes) V and edges E between these nodes
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Lots of applications in many areas: web search,
transportation, biological models, . . .
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Will encounter graphs and graph algorithms in
many different courses
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Combinatorics and Graph Theory
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Example: Social Network as a Graph
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Terminology
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Nodes represent users (Michael,
Jessica, Stuart . . . )
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Two nodes u and v are adjacent if there exists
an edge between them (e.g., nodes 1 and 3)
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Edges represent friendship (e.g.,
Michael is friends with Jessica)
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An edge (u, v ) is incident with nodes u and v
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Edge between nodes u and v is
written as (u, v )
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Degree of a vertex v , written deg(v ), is the
number of edges incident with it
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e.g., (Sarah, Andrew) is an edge in
this graph.
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Neighborhood of a vertex is the set of vertices
adjacent to it
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Question
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Simple Graphs
Consider a graph G with vertices v1 , v2 , v3 , v4 and edges
(v1 , v2 ), (v2 , v3 ), (v1 , v3 ), (v2 , v4 ).
1. Draw this graph.
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Graph contains a loop if any node is adjacent
to itself
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A simple graph does not contain loops and
there exists at most one edge between any
pair of vertices
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Graphs that have multiple edges connecting
two vertices are called multi-graphs
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Most graphs we will look at are simple graphs
2. What is the degree of each vertex?
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Question
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Handshaking Theorem
Let G = (V , E ) be a graph with m edges. Then:
X
deg(v ) = 2m
Consider a simple graph G where two vertices A and B have the
same neighborhood.
v ∈V
Which of the following statements must be true about G?
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Intuition: Each edge contributes two to the sum of the degrees
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Proof: By induction on the number of edges.
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Base case:
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Induction:
A. The degree of each vertex must be even.
B. Both A and B have a degree of 0.
C. There cannot be an edge between A and B .
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Applications of Handshaking Theorem
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Is it possible to construct a graph with 5 vertices where each
vertex has degree 3?
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Prove that every graph has an even number of vertices of odd
degree.
If n people go to a party and everyone shakes everyone else’s
hand, how many handshakes occur?
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Directed Graphs
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All graphs we considered so far are undirected
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In undirected graphs, edge (u, v ) same as (v , u)
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A directed edge (arc) is an ordered pair (u, v )
(i.e., (u, v ) not same as (v , u))
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A directed graph is a graph with directed edges
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In-Degree and Out-Degree of Directed Graphs
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Handshaking Theorem for Directed Graphs
Let G = (V , E ) be a directed graph. Then:
X
X
deg− (v ) =
deg− (v ) = |E |
v ∈V
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The in-degree of a vertex v , written deg− (v ), is the number
of edges going into v
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deg− (a) =
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The out-degree of a vertex v , written deg+ (v ), is the number
of edges leaving v
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deg+ (a) =
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v ∈V
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Subgraphs
P
P
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v ∈V
v ∈V
deg− (v ) =
deg+ (v ) =
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Question
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A graph G = (V , E ) is a subgraph of another graph
G 0 = (V 0 , E 0 ) if V ⊆ V 0 and E ⊆ E 0
Consider a graph G with vertices {(v1 , v2 , v3 , v4 )} and edges
(v1 , v3 ), (v1 , v4 ), (v2 , v3 ).
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Example:
Which of the following are subgraphs of G?
1. Graph G1 with vertex v1 and edge (v1 , v3 )
2. Graph G2 with vertices {v1 , v3 } and edge (v1 , v3 )
3. Graph G3 with vertices {v1 , v3 } and no edges
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4. Graph G4 with vertices {v1 , v2 } and edge (v1 , v2 )
Graph G is a proper subgraph of G 0 if G 6= G 0 .
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Induced Subgraph
Complete Graphs
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Consider a graph G = (V , E ) and a set of vertices V 0 such
that V 0 ⊆ V
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Graph G 0 is the induced subgraph of G with respect to V 0 if:
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A complete graph is a simple undirected graph in which every
pair of vertices is connected by one edge.
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How many edges does a complete graph with n vertices have?
1. G 0 contains exactly those vertices in V 0
2. For all u, v ∈ V 0 , edge (u, v ) ∈ G 0 iff (u, v ) ∈ G
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Subgraph induced by vertices {C , D}:
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Bipartite graphs
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Examples Bipartite and Non-Bi-partite Graphs
A simple undirected graph G = (V , E ) is called bipartite if V
can be partitioned into two disjoint sets V1 and V2 such that
every edge in E connects a V1 vertex to a V2 vertex
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Is this graph bipartite?
A
C
B
A
D
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C
What about this graph?
A
B
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E
E
C
D
B
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Graph Coloring
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A coloring of a graph is the assignment of a
color to each vertex so that no two adjacent
vertices are assigned the same color.
Consider a graph G with vertices {v1 , v2 , v3 , v4 } and edges
(v1 , v2 ), (v1 , v3 ), (v2 , v3 ), (v2 , v4 ).
A graph is k -colorable if it is possible to color it
using k colors.
Which of the following are valid colorings for G?
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e.g., graph on left is 3-colorable
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Is it also 2-colorable?
1. v1 = red, v2 = green, v3 = blue
2. v1 = red, v2 = green, v3 = blue, v4 = red
The chromatic number of a graph is the least number of
colors needed to color it.
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Combinatorics and Graph Theory
Question
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F
3. v1 = red, v2 = green, v3 = red, v4 = blue
What is the chromatic number of this graph?
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Examples
What are the chromatic numbers for these graphs?
A
B
A
B
A
B
C
D
C
D
C
D
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