UCSD CSE 21, Spring 2014
Mathematics for Algorithm and System
Analysis
Week2
Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/
Week 2 Discussion
• UCSD CSE 21, Spring 2014
• Administrivia
– From now on attendance at this discussion section is
counted via clicker questions
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A: I understand.
B: I understand.
C: I understand.
D: I understand.
E: I understand.
Administrivia
• From now on attendance in this discussion is counted
via clicker questions
• Homework Two is due 4/13/2014
• Midterm In-class on May 1 (ABK) and May 2 (RRR)
– 30% of final grade
• This week:
– Lists without repetitions
– Sets
Administrivia
• Personnel changes in CSE21
– I am now covering both Monday sections
– Jay Dessai is no longer a TA for this class
– TUTORS!!!!
• Kacy Raye Espinoza
– krespinoza@ucsd.edu
• Tracy Nham
– tnham@ucsd.edu
• Hours TBD
Review (Theroems / Def’s)
• Cartesian Product : Generalization of Cartesian plane (RxR)
• Lexicographic Order : Generalization of alphabetical order
• Rule of Sum: Size of disjoint union is sum of size of
components
• Rule of Product: Sequence of k choices. The ith choice can
be made in ci ways. Total number of structures is c1 x … x ck
Review (Technique)
• Stars and Bars ( Combinatoric counting method )
– Number 8 from HW1:
– “A monotone increasing number consists of digits taken from the set {1,
2, …, 9}, with each digit greater than or equal to its neighbor digit to the
left (if that digit exists). E.g., 1112256888899 is a monotone increasing
number with 13 digits. How many 6-digit monotone increasing numbers
are there? ”
– Applicable Theorem:
– For any pair of natural numbers n and k, the number of distinct n-tuples
of non-negative integers whose sum is k is given by the binomial
coefficient 𝑛+𝑘−1
𝑘
Review (Technique)
• Stars and Bars ( Combinatoric counting method )
– Number 8 from HW1:
– Applicable Theorem:
– For any pair of natural numbers n and k, the number of distinct n-tuples
of non-negative integers whose sum is k is given by the binomial
coefficient 𝑛+𝑘−1
𝑘
– The things we’re actually counting are not actually {1,2,…,9}
– They’re stars and bars!
Review (Technique)
• Stars and Bars ( Combinatoric counting method )
– Number 8 from HW1:
– Answer is
9+6−1
6
– Why is k = 6 ?
– k = 6 because there are 6 – 1 = 5 divisions between the digits
– n = 9 because we have 9 possible items
Subsets
• Example: Consider set S = { x, y, z }
– How many 2-lists does S generate?
• A: 3
• B: 6
• C: 9
• D: 8
• E: 0
Subsets
• Example: Consider set S = { x, y, z }
– How many 2-lists does S generate?
• A: 3
• B: 6
• C: 9
• D: 8
• E: 0
Subsets
• Example: Consider set S = { x, y, z }
– How many 2-lists without repetitions?
• A: 3
• B: 6
• C: 9
• D: 8
• E: 0
Subsets
• Example: Consider set S = { x, y, z }
– How many 2-lists without repetitions?
• A: 3
• B: 6
• C: 9
• D: 8
• E: 0
Subsets
• Example: Consider set S = { x, y, z }
– How many 2-sets which are subsets?
• A: 3
• B: 6
• C: 9
• D: 8
• E: 0
Subsets
• Example: Consider set S = { x, y, z }
– How many 2-sets which are subsets?
• A: 3
• B: 6
• C: 9
• D: 8
• E: 0
Subsets
• Example: Consider set S = { x, y, z }
– 2-lists: there are 32 = 9
– 2-lists without repetitions: 3*2 = 6
– 2-sets which are subsets: ??? How many???
» { x, y }
{ x, z }
{ y, z }
Theorem 7: k-subsets of an n-set
• Proof: Each k-subset is the set of elements of k! klists without repetitions!
Up Next: Probability!
• Counting and Probability go hand in hand
• Here is a game that demonstrates this