CSE390D Notes for Monday, 11/4/24
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Basics of Counting:
sum rule
product rule
overcounting (subtraction rule, principle of inclusion-exclusion)
decision trees
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product rule: outcome is a sequence of outcomes from various sets
outcome = A1 x A2 x A3 ... x An
total = |A1| * |A2| * |A3| ... * |An|
example: one of 5 appetizers AND one of 8 entries AND one of 6 desserts
total? 5 * 8 * 6
license plates: 3 letters followed by 3 digits
answer: 26^3 * 10^3
38 students in a class, 38 chairs, how many arrangements?
38!
functions from a set of n elements to a set of m elements?
m^n
one-to-one functions from a set of n elements to a set of m elements?
m * (m-1) * (m-2)... * (m - n +1) = m! / (m-n)!
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sum rule: outcome is one of a set of disjoint sets
outcome = {A1 U A2 U A3 ... U An}
total = |A1| + |A2| + |A3| ... + |An|
example: one of 3 fish entries OR 2 veggie entries OR 4 meat entries
total? 3 + 2 + 4
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BASIC variable names
variables can be one or two characters long
* The first character must be a letter
* The second character can be a letter or a digit
* The keywords “TO”, “IF”, and “DO” are excluded
answer: 26 + 26 * 36 - 3 = 26 * 37 - 3
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passwords are 4-6 chars, at least one digit and at least one letter
(case insensitive, no special characters):
Answer:
(4-char pass) = 36^4 (all strings) - 26^4 (no digit) - 10^4 (no letter)
(5-char pass) = 36^5 (all strings) - 26^5 (no digit) - 10^5 (no letter)
(6-char pass) = 36^6 (all strings) - 26^6 (no digit) - 10^6 (no letter)
overall answer is the sum: (4-char pass) + (5-char pass) + (6-char pass)
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subtraction rule: outcome is one of a set of overlapping sets
outcome = A1 U A2
total = |A1| + |A2| - |A1 intersect A2|
also called the principle of exclusion-inclusion
we overcount, then compensate
How many two-digit numbers that begin with even or end with even?
(# begin) + (# end) - (# begin & end) =
4 * 10 + 9 * 5 - 4 * 5 = 40 + 45 - 20 = 65
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others:
subsets of set of n elements?
2^n
binary digits of length 8 start with 1 or end with 00?
2^7 + 2^6 - 2^5
class has 40 students, 20 CS, 15 math, 5 dual
how many neither math nor cs?
#cs/math = 20 + 15 - 5 = 30
non cs/math = 10
best 2 out of 3 for joe/sally...how many outcomes? 6
ss, jj, sjs, jss, jsj, sjs
use a tree diagram
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Stuart Reges
Last modified: Mon Nov 4 13:30:02 PST 2024