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Counting Principles and Permutations
A great way to begin a discussion about permutations is to read the book, "Annos Mysterious Multiplying Jar". Ask the students to keep track of the "items" as you read the story. Then, discuss when order makes a difference. For example, there are three bedrooms that you and your siblings will choose from, one that has everything, the second with the basics, and the third is a paper box. Students see the importance of order. Have students come up with their own problems where order makes a difference. For example, being chosen for a baseball team as opposed to the batting order of a team.
by Sally Weaver <weaves@pacific.net>
http://forum.swarthmore.edu/t2t/discuss/message.taco?thread=3398&n=1
from the Teacher@Teacher Public Discussion at the Math Forum
Assignment Exercises
Readings Bello & Britton,
Sections 9.1-9.2Getting Beyond the Information
from the
Center for Teaching and Learning
at Indiana State University (http://web.indstate.edu:80/ctl/
progr/Tips/tip10.html)
Web Resources These web resources have links to counting principles, factorials, and permutations.
Please use a calculator with a factorial key for the problems this week.
For permutations problems, use the permutation key on your calculator or this little permutation calculator found at:
http://www.topangasoftware.com/PermutationCalculator.htm
or the factorial calculator at:
http://www.ttu.ee/it/vorgutarkvara/wai4040/KjellJavaLessons/Notes/chap17/ch17_11.html
1) Look over the reading this week and then in reference to "Getting Beyond the Information," describe
a) a short activity that shows how you introduce counting principles in your classOR
b) a short example from your life where you would use a counting principle or a permutation.
2) Complete the following exercises.
Exercises
...
1) pp. 602-605, Ex. 2, 4, 7, 8, 11, 13, 23, 37, 42 43
...
2) pp., 612- 615, ex. 2, 3, 4, 8, 10, 14, 16, 23, 24, 25, 29, 30, 41, 43, 45, 46, 53
...
3) One problem with permutations is if the items to be permutated have objects that repeat. For example, CHEESE would have 6! ways of arranging itself. But the problem is that CHEESE is the same of CHEESE when the first and second "E's" are switched. And there are 3! ways of interchanging those "E's," since there are 3 of them. So to make sure that different arrangements are found in permutations which have some like objects,, divide by the factorial of the number of the like objects. For example, how many distinct 6 letter words can be made from KANSAS would be:
6!/(2! * 2!) = 180.How many different words can be made using all the letters from:
a) WISCONSIN
b) MISSISSIPPI...
4) Tom has 12 poker chips; 5 are yellow, 4 are red, and 3 are blue. In how many different-looking stacks of 12 chips can he pile them?
...
5) A coin is to be tossed 10 times. How many possible results are there? HINT: Think of this as a making 10-letter words using H's and T's.
...
6) One of the ways people devise codes is to take a word - such as CAB and from all of the one, two, and three letter words possible. they then match those :nonsense" words with a letter. For CAB, there are 6 three letter words, 6 two letters words and 3 1 one letter words, so that would only give 15 "nonsense" words to match with the letters of the alphabet which his not enough. How many code words can be formed using the the letters from the word: CART?
...
7) Simplify:
a)
b)
Ask Dr. Math about Factorials
http://mathforum.org/dr.math/tocs/factorial.middle.htmlAsk Dr. Math about The Three Canteens - a great example of a tree diagram
http://mathforum.org/dr.math/problems/joe9.18.97.htmlCombinatorics: Let me Count the Ways - Lessons and Projects with Math and Literature - The literature links do not work, but the ideas are excellent~
http://www.kqed.org/ednet/school/math/mathonline/lessons/combinatorics/Introduction to Factorials - from Math Tables Facts and Formulas, Hoxie High School, Math Department
http://24.225.17.65/math/count/facto.htmMath Lesson Plans on Permutations - by Intern teacher Cristal Glass, University of Saskatchewan, Canada with lots of handouts, examples, and exercises
http://www.usask.ca/education/ideas/tplan/mathlp/math.htmPermutations and Combinations Introduction - from the Math Forum
http://mathforum.org/dr.math/faq/faq.comb.perm.htmlRoger Day's course material from Illinois State University on Counting Principles
(http://www.math.ilstu.edu/%7Eday/courses/old/312/session17.html)
and on Permutations
(http://www.math.ilstu.edu/%7Eday/courses/old/312/session18.html)The (Combinatorial) Object Server - will generate permutations and seems more complicated than it is. Just change the number after "n =" and then click generate to receive a list of all the possible permutations of a set of length n!
http://www.theory.csc.uvic.ca/~cos/gen/perm.htmlAn Internet Lesson using Permutations
How Many Ways Can A Team Win A 7-Game Series
http://score.kings.k12.ca.us/lessons/teamwins.htm
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