1 Binomial Coefficient Supplementary Notes Prepared by Raymond Wong Presented by Raymond Wong
Binomial Coefficient
Jan 14, 2016
Binomial Coefficient. Supplementary Notes. Prepared by Raymond Wong. Presented by Raymond Wong. e.g.1 (Page 4). Prove that. S 1. S 2. S 3. S 4. S 0. {1}. {1, 2, 3}. {1, 2}. {1, 3}. {2}. {1, 2, 3, 4}. {1, 2, 4}. {1, 4}. {2, 3}. {}. {3}. {1, 3, 4}. {2, 4}. {3, 4}. {4}. - PowerPoint PPT Presentation
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1
Binomial Coefficient
Supplementary Notes
Prepared by Raymond WongPresented by Raymond Wong
2
e.g.1 (Page 4) Prove that
44
0
24
i i
44
0
24
i i
3
e.g.1A set of all possible subsets of {1, 2, 3, 4}
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
44
0
24
i i
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
S0
S1 S2 S3
S4
4
e.g.1A set of all possible subsets of {1, 2, 3, 4}
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
S0
S1 S2 S3
S4
40
41
42
43
44
44
0
24
i i
+ + + +
5
e.g.1A set of all possible subsets of {1, 2, 3, 4}
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
44
0
24
i i
{2}
1 does not appear2 appears3 does not appear4 does not appear
{2, 3, 4}
1 does not appear2 appears3 appears4 appears
We can have another representation (related to “whether an element appears or not”)to represent a subset
6
e.g.1A set of all possible subsets of {1, 2, 3, 4}
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
44
0
24
i i
1
Appears
Does notappear
2
Appears
Does notappear
3
Appears
Does notappear
4
Appears
Does notappear
{2}
7
e.g.1A set of all possible subsets of {1, 2, 3, 4}
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
44
0
24
i i
1
Appears
Does notappear
2
Appears
Does notappear
3
Appears
Does notappear
4
Appears
Does notappear
{2, 3, 4}
8
e.g.1A set of all possible subsets of {1, 2, 3, 4}
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 4}
{2, 4}
{1, 3}
{2, 3}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
44
0
24
i i
1
Appears
Does notappear
2
Appears
Does notappear
3
Appears
Does notappear
4
Appears
Does notappear
2 choices 2 choices 2 choices 2 choices
Total number of subsets of {1, 2, 3, 4} = 2 x 2 x 2 x 2 = 24
9
e.g.2 (Page 16) Prove that
2
4
1
4
2
5
2
4
1
4
2
5
10
e.g.2
2
4
1
4
2
5
S
A
B
C
D
E
{A, B}
{A, C}
{A, D}
{B, C}
{B, D}
{C, D}
{A, E}
{B, E}
{C, E}
{D, E}
A set of all possible 2-subsets of S
52
11
e.g.2
2
4
1
4
2
5
S
A
B
C
D
E
{A, B}
{A, C}
{A, D}
{B, C}
{B, D}
{C, D}
{A, E}
{B, E}
{C, E}
{D, E}
A set of all possible 2-subsets of S
{A, B}
{A, C}
{A, D}
{B, C}
{B, D}
{C, D}
A set of all possible 2-subsets of S notcontaining E
{A, E}
{B, E}
{C, E}
{D, E}
A set of all possible 2-subsets of S containing E
12
e.g.2
2
4
1
4
2
5
S
A
B
C
D
E
{A, B}
{A, C}
{A, D}
{B, C}
{B, D}
{C, D}
A set of all possible 2-subsets of S notcontaining E
{A, E}
{B, E}
{C, E}
{D, E}
A set of all possible 2-subsets of S containing E
A set of all possible 2-subsets of {A, B, C, D} S’
A
B
C
D
42
We know that each 2-subset contains E.Since each 2-subset contains 2 elements, the other ONE element comes from {A, B, C, D} S’
A
B
C
D
41
+
This proof is an example of a combinatorial proof.
13
e.g.3 (Page 22) Prove that
32233
3
3
2
3
1
3
0
3)( yxyyxxyx
32233
3
3
2
3
1
3
0
3)( yxyyxxyx
14
e.g.3
32233
3
3
2
3
1
3
0
3)( yxyyxxyx
x
y
blue
x
y
red
x
y
green
monomial
15
e.g.3
32233
3
3
2
3
1
3
0
3)( yxyyxxyx
y
Set of y’s (in different colors)
y
y
Coefficient of xy2=No. of ways of choosing 2 y’s from this set32=
Two y’s are in different colors.
Interpretation 1
Suppose that we choose 2 elements.These two elements correspond to two y’s indifferent colors.
16
e.g.3
32233
3
3
2
3
1
3
0
3)( yxyyxxyx
Coefficient of xy2=No. of lists containing 2 y’s 32=
L1 L2 L3 Each Li can be x or y.Now, I want to have 2 y’s in this list. 1
Set of positions in the list
2
3Suppose that we choose 2 positions from the set.These two positions correspond to the positions thaty appears.
e.g., {1, 3} means yxy
Each monomial has 3 elements.Each element can be x or y.
Interpretation 2
These 2 y’s appear in 2 different positions in this list.
17
e.g.3
32233
3
3
2
3
1
3
0
3)( yxyyxxyx
Coefficient of xy2=
L1 L2 L3
1
Set of positions in the list
2
3
Interpretation 3
18
e.g.3
32233
3
3
2
3
1
3
0
3)( yxyyxxyx
Coefficient of xy2=No. of ways of distributing 3 objects 32=
L1 L2 L3
1
Set of positions in the list
2
3 Choose 2 objects
Bucket B132
Bucket B2
Interpretation 3
19
e.g.4 (Page 29)
Suppose we have 2 distinguishable buckets, namely B1 and B2
How many ways can we distribute 5 objects into these buckets such that 2 objects are in B1
3 objects are in B2?
20
e.g.4
Choose 2 objects
B1
B2
52
21
e.g.5 (Page 30)
Suppose we have 3 distinguishable buckets, namely B1, B2 and B3.
How many ways can we distribute 9 objects into these buckets such that 2 objects are in B1
3 objects are in B2, and 4 objects are in B3?
9 objects 2 objects in B1, 3 objects in B2 and 4 objects in B3
22
e.g.5
Choose 2 objects
B1
92
9 objects 2 objects in B1, 3 objects in B2 and 4 objects in B3
23
e.g.5
Choose 2 objects
B1
92
Choose 3 objects
B273
B3
Total number of placing objects = 92
73
x9!
2!(9-2)! 3!(7-3)!
7!x=
9!
2!3!4!=
9 objects 2 objects in B1, 3 objects in B2 and 4 objects in B3
24
e.g.6 (Page 33) Prove that
the coefficient of in (x + y + z)4
is equal to
where k1 + k2 + k3 = 4
321 kkk zyx
321
4
kkk
25
e.g.6
Coefficient of x2yz= 42 1 1
=
L1 L2 L3
L4
Choose 2 objects
Bucket B1
Bucket B2
(x + y + z) (x + y + z) (x + y + z) (x + y + z)= xxxx + xxxy + xxxz + xxyx + … + zzzy + zzzz
1
Set of positions in the list
2
3
4
Choose 1 object Bucket B3
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