# Probability 1: Combinatorial Analysis

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 Author: barium ID: 220740 Filename: Probability 1: Combinatorial Analysis Updated: 2013-05-28 21:07:15 Tags: statistics probability Folders: Description: Combinatorial analysis formulas and definitions used in probability theory Show Answers:

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1. Define combinatorial analysis (or combinatorics)
Combinatorics is a branch of mathematics that studies countable discrete structures. It includes studying counting, ordering and other problems.
2. The basic principle of counting
If event A has m possible outcomes, and event B has n possible outcomes, then there are m*n possible outcomes of the two experiments together.
3. Number of permutations of n objects
n*(n-1)*(n-2)*...*3*2*1 = n!
4. Number of permutations of n objects, of which n1 are alike, n2 are alike, ... nr are alike.
5. The number of distinct subsets of size k that can be selected from a set of n objects (order of objects is irrelevant)
, a.k.a. the binomial coefficient.
6. The number of distinct ordered subsets of size k that can be selected from a set of n objects (order of objects is relevant)
7. Pascal's Rule (a combinatorial identity about binomial coefficients)
, for 1 ≤ r ≤ n
8. The binomial theorem
• Binomial theorem describes the algebraic expansion of powers of a binomial.
9. The number of subsets of a set of n elements
2n   (this includes the null subset)
10. The number of possible partitions of a set of n objects into r distinct groups (order irrelevant)
, a.k.a. the multinomial coefficient
11. The multinomial theorem
•
• (the sum is over all non-negative integers n1, n2, ..., nr such that n1+n2+...+n= n)
12. The number of distinct positive integer-valued vectors  satisfying
13. The number of distinct non-negative integer-valued vectors  satisfying
14. Type I error
• A false positive
• The null hypothesis is rejected when it is actually true.
• "I falsely think that the Alternative hypothesis is true."
15. Type II error
• A false negative
• The null hypothesis is accepted when it is actually false.
• "I falsely think that the Alternative hypothesis is false."

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