1 / 10

Combinatorial Principles, Permutations, and Combinations

Section 03. Combinatorial Principles, Permutations, and Combinations. Permutations vs. Combinations. Permutations are ordered Combinations are not ordered Therefore, there are more permutations than combinations for given and Both apply to combinatorics without replacement

abbott
Download Presentation

Combinatorial Principles, Permutations, and Combinations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 03 Combinatorial Principles, Permutations, and Combinations

  2. Permutations vs. Combinations • Permutations are ordered • Combinations are not ordered • Therefore, there are more permutations than combinations for given and • Both apply to combinatorics without replacement • Also: remember 0! = 1

  3. Permutations • ORDER MATTERS • Choosing an ordered subset of size from a collection of objects without replacement: • Given objects, of which are Type 1, are Type 2, etc, up to , the number of ways to order all objects is:

  4. Combinations • ORDER DOES NOT MATTER • Choosing a subset of size from a collection of objects without replacement: • is also called a binomial coefficient

  5. Binomial Theorem • is called the binomial coefficient because it is used in the power series expansion of • If N is as integer, summations stops at k=N • If N is not an integer, series is only valid if -1<t<1 • This expansion is useful for understanding the binomial distribution

  6. Multinomial Theorem • In the power series expansion of the coefficient of is • Useful for understanding multinomial distributions (much later)

  7. In conclusion… • Ordered? • Not ordered?

  8. Actex, Sec 3 #1, pg 103 A class contains 8 boys and 7 girls. The teacher selects 3 of the children at random and without replacement. Calculate the probability that the number of boys selected exceeds the number of girls selected.

  9. Actex, Sec 3, #3, pg 103 A box contains 4 red balls and 6 white balls. A sample of size 3 is drawn without replacement from the box. What is the probability of obtaining 1 red ball and 2 white balls, given that at least 2 of the balls in the sample are white?

  10. Sample Exam #16 An insurance company determines that , the number of claims received in a week, is a random variable with The company also determines that the number of claims received in a given week is independent of the number of claims received in any other week. Determine the probability that exactly seven claims will be received during a given two-week period.

More Related