Classroom Activity-Discrete Mathematics

Combinations- How many ways can you…………

 

Resources Needed: 2-5 standard decks of cards, a computer with Internet access and possibly a LCD projector, Microsoft Excel for the combination calculations, any Texas Instrument calculator (TI-82, TI-83, TI-85, TI-86, or TI-92), and informational handouts.

Lesson Activity

Exposition:

  1. Begin the lesson with information and handouts on discrete mathematics.
  2. Discuss the importance of combinatorics and the counting of finite sets.
  3. Display examples of combinations and permutations- how computer speed is determined by the complexity of algorithms, how the demand for phone numbers and Internet addresses are met using all types of different combinations, and how counting techniques are used to determine probability in games of chance.
  4. Use a standard deck of cards to show students techniques necessary to obtain a red picture card. With these different techniques, students will complete a lab worksheet.

Experience:
  1. Instructor will demonstrate how cards can be chosen from a standard deck of cards.
  2. Briefly explain combinations and permutations.
  3. Students will use the computer to go to the Internet site of Elsevier Science. They will explore recent written literature that pertains to the field of combinatorics.
  4. Students will use the Internet to research the site http://www.combinatorics.net/. In groups of 2 or 3 they will then chose either permutations or combinations to make up questions for the in-class assignment.

Critical Thinking:
  1. Why is the field of combinatorics (permutations and combinations for our purpose) so important for a network of computers? Think of two roles that permutations and combinations (one for each) play in determining the probability of a certain event. Discuss the differences between permutations and combinations.

Interactive Learning:
  1. Students will use the decks of cards to complete an in class lab worksheet on determining the probability of how different hands of poker can be obtained.

Assignment:
  1. Each student, outside of class, will write a 1 page summary (on info from) two additional Internet sites on permutations or combinations, and include why these two sites would be beneficial to research for someone who wants to get more information on permutations or combinations.

Resources:

HTTP://www.elsevier.com/inca/publications/store/5/0/5/6/1/0/505610.relpub.htt

HTTP://www.combinatorics.net/

HTTP://www.geocities.com/Athens/Delphi/5136/Induction.html

HTTP://vig.prenhall.com/refbook/0,2593,0137275870,00.html

HTTP://www.usask.ca/education/ideas/tplan/mathlp/math.htm

 

 

Name:_________________

Permutations and Combinations Laboratory

  1. List all the permutations of {a, b, c}

    2.  

     

  2. How many permutations are there of the set {a, b, c, d, e, f, g}?

    3.  

     

  3. How many permutations of {a, b, c, d, e, f, g} end with a?

    4.  

     

  4. Let S = {1, 2, 3, 4, 5}

    1. How many 3-permutations of S are there?

      2.  

       

    2. How many 3-combinations of S are there?

     

     

     

  5. How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible?

 

 

 

Answers:

  1. {a, b, c}, {a, c, b}, {b, a, c}, {b, c, a}, {c, a, b}, {c, b, a}

  2. 7! = 5040

  3. 6! = 720

    1. P(5,3) = 5! / (5-3)! = 60

    2. C(5,3) = 5! / (3!(5-3)!) = 10

  5. 12* 11* 10 = 1320