L2 Supplementary Notes
Page 1
09-02-2010: Recap
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General Product Principle
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Counting Functions
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Functions are lists
Function concepts
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For counting list
Injection, surjection, and bijection
Bijiection Principle
L2 Supplementary Notes
09-02-2010: Product Principle
Page 2
L2 Supplementary Notes
09-02-2010: Recap/Counting Functions
Page 3
L2 Supplementary Notes
09-02-2010: Injection and Surjection
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L2 Supplementary Notes
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09-02-2010: Bijection
L2 Supplementary Notes
09-02-2010: Bijection Principle
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L2 Supplementary Notes
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09-02-2010: Today
L2 Supplementary Notes
Page 8
Increasing Triples
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Can we count using general product principle?
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# of possible values for i?
 n-2
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# of possible values for j?
 Depends on value of i!
 Violates condition of general product principle
L2 Supplementary Notes
Page 9
Use Bijection Principle
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n=4
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All increasing triples listed.
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Do we have all the subsets? How about subset {2, 1, 3}?
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It is the same as {1, 2, 3}
L2 Supplementary Notes
Page 10
Increasing Triples
L2 Supplementary Notes
Page 11
Increasing Triples
L2 Supplementary Notes
Page 12
Increasing Triples
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Same as
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The number of 3-element subsets from {1, 2, …, n}
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What is this number?
First consider another number
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The number of 3-element permutations from {1, 2, …, n}
L2 Supplementary Notes
 Order matters, not required to be increasing
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3-element permutations from {1, 2, 3, 4}
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L2 Supplementary Notes
General product principle can be applied here
although cannot for increasing triples. Why?
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Page 14
L2 Supplementary Notes
3-element subsets/3-element permutations
Page 15
L2 Supplementary Notes
3-element subsets/Increasing triples
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Same as
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The number of 3-element subsets from {1, 2, …, n}
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L2 Supplementary Notes
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L2 Supplementary Notes
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Permutations
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Number of k-element permutations
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Number of permutations of a set of size n
L2 Supplementary Notes
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K-th falling factorial
L2 Supplementary Notes
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L2 Supplementary Notes
k-element subsets/k-elemen permutations
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L2 Supplementary Notes
k-element subsets/k-elemen permutations
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L2 Supplementary Notes
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Summary
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Number of increasing triples:
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Number of 3-element subsets
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Number of 3-element permutations
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Number of k-element subsets
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Number of k-element permutations
L2 Supplementary Notes
Page 25
Assignment 2: Problem 9
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Exco Members: Year 1: 4; Year 2: 5; Year 3: 3
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Total number of members: 12
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Choose 6 from 12
L2 Supplementary Notes
Page 26
Assignment 2: Problem 9
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Exco Members: Year 1: 4; Year 2: 5; Year 3: 3
Number of ways to choose Treasury Subcommittee
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12 choose 3
Number of ways to choose Events Subcommittee
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9 choose 3
L2 Supplementary Notes
Page 27
Assignment 2: Problem 9
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Exco Members: Year 1: 4; Year 2: 5; Year 3: 3
Number of ways to choose Treasury Subcommittee
Number of ways to choose Events Subcommittee
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9 choose 3
Answer
L2 Supplementary Notes
Page 28
Assignment 2: Problem 9
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Exco Members: Year 1: 4; Year 2: 5; Year 3: 3
“At least” makes things complex.
L2 Supplementary Notes
L2 Supplementary Notes
Wrong Answer
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First choose 1 from each year
Then pick 3 from remaining 9 members
Answer
L2 Supplementary Notes
Wrong Answer
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First choose 1 from each year
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Then pick 3 from remaining 9 members
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Anthony (Y1), Mike (Y2), William (Y3)
Billy (Y1), Calvin(Y1), David (Y1),
Committee
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Anthony (Y1), Billy (Y1), Calvin(Y1), David (Y1), Mike (Y2), William (Y3)
L2 Supplementary Notes
Wrong Answer
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First choose 1 from each year
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Then pick 3 from remaining 9 members
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Anthony (Y1), Calvin(Y1), David (Y1),
Same Committee as before
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Billy (Y1), Mike (Y2), William (Y3)
Anthony (Y1), Billy (Y1), Calvin(Y1), David (Y1), Mike (Y2), William (Y3)
The same committee counted twice.
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Not a bijection