Unit 6

Permutations, Combinations, and The Binomial Theorem

Investigation

This Lesson deals with counting objects. You can do this by writing out every possibility, but sometimes listing all possibilities can be time-consuming, not to mention prone to error. Noticing patterns will often allow you to count indirectly. The following activity shows an example of this.

Suppose you are playing a game, and you are holding «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» playing cards: J, Q, K, and A.

  1. In how many different ways can you play one card from your hand of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math»?

  1. Suppose you need to play two cards, where the order of play matters (that is, playing JQ is different than playing QJ). How many different two-card plays can you make if

    a.  repetition of cards is allowed?
    b.  repetition of cards is not allowed?

  1. Now, suppose you need to play «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» cards, where the order of play matters. How many different three-card plays can you make if

    a.  repetition of cards is allowed?
    b.  repetition of cards is not allowed?

  1. How many four-card plays can you make, where order matters, if

    a.  repetition of cards is allowed? (Do you really want to continue with that tree diagram?)
    b.  repetition of cards is not allowed?

  1. Now, suppose you have a complete deck of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»52«/mn»«/mstyle»«/math» cards. Use patterns from the previous questions to write an expression showing the number of ways you could play «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»52«/mn»«/mstyle»«/math» cards, where order matters, if

    a.  repetition of cards is allowed?
    b.  repetition of cards is not allowed?
Tree diagrams for questions 2a and 2b have been started for you.



  1. «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math»

  1. a.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»16«/mn»«/mstyle»«/math»
    b.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»12«/mn»«/mstyle»«/math»

  1. a.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»64«/mn»«/mstyle»«/math»
    b.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»24«/mn»«/mstyle»«/math»

  1. a.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»256«/mn»«/mstyle»«/math»
    b.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»24«/mn»«/mstyle»«/math»

  1. a.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»52«/mn»«mn»52«/mn»«/msup»«/mstyle»«/math»
    b.  «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»52«/mn»«mo»§#215;«/mo»«mn»51«/mn»«mo»§#215;«/mo»«mn»50«/mn»«mo»§#215;«/mo»«mo»§#8230;«/mo»«mo»§#215;«/mo»«mn»3«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math»
If you were unable to confidently complete this activity, contact your teacher.