1 / 14

Section 9-1

Section 9-1. Counting Techniques. Section 9-1. counting multiplication principle of counting permutations special permutation situations combinations subsets. Counting.

noel-richmond
Download Presentation

Section 9-1

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 9-1 Counting Techniques

  2. Section 9-1 • counting • multiplication principle of counting • permutations • special permutation situations • combinations • subsets

  3. Counting • finding the number of different ways a particular “experiment” can be completed or the number of different arrangements of objects • the simplest counting technique is to list all of the possible outcomes • sometimes a tree diagram can be used to show the different outcomes

  4. Multiplication Principle of Counting • sometimes listing the different outcomes or making a tree diagram would be too time consuming • the multiplication principle of counting (MPC) is a shortcut to algebraically do the same thing as a list or diagram • MPC – if a procedure can be broken into stages, find the number of ways each stage can be completed and then take the product these results

  5. M.P.C. Find the number of different license plates that can be made consisting of three letters followed by three digits

  6. M.P.C. Find the number of different license plates that can be made consisting of three letters followed by three digits L L L D D D

  7. M.P.C. Find the number of different license plates that can be made consisting of three letters followed by three digits L L L D D D 26 26 26 10 10 10 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000 ways

  8. M.P.C. If the same problem was done but the characters could not repeat, the answer would change L L L D D D 26 25 24 10 9 8 26 x 25 x 24 x 10 x 9 x 8 = 11,232,,000 ways

  9. Permutations • a special application of MCP is to look at how many different ways that a set of n objects can be arranged in order • each such ordering is called a permutation • formula: the number of permutations of n objects is n! • n! (read as “n factorial”) represents the product

  10. Special Permutations • if the n objects are not all distinguishable, then a special formula must be used to divide away the “repeated” arrangements • formula: • each object that repeats needs to be taken care of in the denominator, with n1, n2, etc. being how many there are of each

  11. Special Permutations • in many counting problems, we are interested in using n objects, but only filling r blanks • this can be done with MPC or a special formula • the number of permutations of n objects taken r at a time is:

  12. Combinations • in many applications, we are interested in ways to select r objects from a group of n objects but the order of the arrangement doesn’t matter • these unordered selections are called “combinations” • the number of combinations of n objects taken r at a time is:

  13. Permutations vs. Combinations • when completing problems it is important identify whether for different selections order matters (permutations) or doesn’t matter (combinations) • each of these can be used in conjunction with the MPC to figure out how many ways each stage can be done (although only permutations can be done with MPC) • since a permutation is the ordering of a certain selection of objects:

  14. Subsets • a different kind of counting problem involves the number of subsets of a set of n objects • the formula for this comes from MPC; each of the n objects has 2 possibilities, either it is included or not included; break the selection into n stages each having 2 ways it can be completed • this leads to the formula for the number of subsets of n objects:

More Related