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CS245 Exam 4
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Terms in this set (20)
Generalized pigeonhole Principle
if n items are placed in k boxes, then at least one box contains at least ⌈n/k⌉ items.
Multiplication Principle (Product Rule)
If there are s steps in an activity, with n1 ways to accomplish the first step, n2 ways to accomplish the second and ns ways to accomplish the last step, then there are n1
n2
...* ns ways to accomplish all steps.
Addition Principle (Sum Rule)
If there are t tasks, with n1 ways to accomplish the first task, n2 ways to accomplish the second and nt ways to accomplish the last task, then there are n1 + n2 +...+ nt ways to complete 1 task.
Principle of Inclusion Exclusion of 2 sets
the cardinality of the union of sets M and N is the sum of their individual cardinalities excluding the cardinality of their intersection.
Principle of Inclusion Exclusion of 3 sets
The cardinality of the union of sets M, N, and O is the sum of their individual cardinalities excluding the sum of the cardinalities of their pairwise intersections and including the cardinality of their intersection
Permutation
An ordering of n distinct elements
r - Permutation
An ordering of a r element subset of n distinct elements
r Combination
an r element subset of an n element set. The number of subsets is denoted C(n, r)
Combinatorial proof
An argument based on the principles of counting
Algorithm
A finite set of instructions for performing a task
Recursive definition
Has 3 parts, the basis clause, the inductive clause, the extremal clause
Basis clause
Determines how trivial cases are handled
Inductive clause
Explains how complex problems are answered in terms of simpler
Extremal clause
Only cases covered by the basis and inductive clauses are covered by the recursive definition. That is this provides boundaries for the definition.
Recursive Algorithm
Expresses the solution to a task in terms of a simpler case of the same problem.
Factorial
The ___ of a non-negative integer n, denoted n! is the product of all integer values from 1 through n, inclusive. By definition 0! = 1
Fibonacci Sequence
In which the nth term is the sum of terms n-1 and n-2 where F(0) = 0 and F(1)=1
Recurrence relations
for the sequence a0, a1, ... is an equation that expresses term ak in terms of one or more of its preceding sequence members, one or more of which are explicitly stated initial conditions of the sequence.
linear homogeneous recurrence relation with constant coefficients of degree k
Has the form R(n) = c1R(n-1) + c2R(n-2) + ... + ckR(n-k) where ci in R and ck does not equal 0
Probability
The ratio of the number of occurrences of interest to the number of possible occurrences.
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