32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL March 12-14 ictcm.com | #ICTCM
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM You Can “Count” on Me! A Counting Strategy for Undergraduates
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM Dr. Scott Demsky Associate Professor Broward College Department of Mathematics Davie, Florida sdemsky@broward.edu
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM You Can “Count” on Me! A Counting Strategy for Undergraduates Students in Liberal Arts Math, Quantitative Reasoning or • Statistics often have difficulty counting the number of ways in which an event can occur.
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM You Can “Count” on Me! A Counting Strategy for Undergraduates Students in Liberal Arts Math, Quantitative Reasoning or • Statistics often have difficulty counting the number of ways in which an event can occur. Should they use: • – The Fundamental Counting Principle?
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM You Can “Count” on Me! A Counting Strategy for Undergraduates Students in Liberal Arts Math, Quantitative Reasoning or • Statistics often have difficulty counting the number of ways in which an event can occur. Should they use: • – The Fundamental Counting Principle? – Factorials or Permutations?
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM You Can “Count” on Me! A Counting Strategy for Undergraduates Students in Liberal Arts Math, Quantitative Reasoning or • Statistics often have difficulty counting the number of ways in which an event can occur. Should they use: • – The Fundamental Counting Principle? – Factorials or Permutations? – Combinations?
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM You Can “Count” on Me! A Counting Strategy for Undergraduates Students in Liberal Arts Math, Quantitative Reasoning or • Statistics often have difficulty counting the number of ways in which an event can occur. Should they use: • – The Fundamental Counting Principle? – Factorials or Permutations? – Combinations? – More than one of these?
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM You Can “Count” on Me! A Counting Strategy for Undergraduates Students in Liberal Arts Math, Quantitative Reasoning or • Statistics often have difficulty counting the number of ways in which an event can occur. Should they use: • – The Fundamental Counting Principle? – Factorials or Permutations? – Combinations? – More than one of these? We will see how the following one-page decision chart • can be used to help students learn how to “count” successfully.
32 nd International Conference on Dr. Scott H. Demsky Department of Mathematics Technology in Collegiate Mathematics Broward College, Davie, Florida ORLANDO, FL | MARCH 12-14 sdemsky@broward.edu #ICTCM You Can “Count” on Me! Are the objects Yes you’re counting all A Counting Strategy for Undergraduates the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No Use Use Factorial: n! Permutations: nPr
Let’s try some examples!
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM Don has 5 pairs of shoes, 2 pairs of pants, and 3 shirts. If all items match, how many different outfits can he wear?
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM Don has 5 pairs of shoes, 2 pairs of pants, and 3 shirts. Are the objects If all items match, how many different outfits can he wear? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM Don has 5 pairs of shoes, 2 pairs of pants, and 3 shirts. Are the objects If all items match, how many different outfits can he wear? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM Don has 5 pairs of shoes, 2 pairs of pants, and 3 shirts. Are the objects If all items match, how many different outfits can he wear? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No 5 ∙ 2 ∙ 3 = 30 Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM A multiple-choice test has 6 questions, each with 4 possible answers. How many ways are there to mark the answers?
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM A multiple-choice test has 6 questions, each with 4 possible Are the objects answers. How many ways are there to mark the answers? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM A multiple-choice test has 6 questions, each with 4 possible Are the objects answers. How many ways are there to mark the answers? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM A multiple-choice test has 6 questions, each with 4 possible Are the objects answers. How many ways are there to mark the answers? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM A multiple-choice test has 6 questions, each with 4 possible Are the objects answers. How many ways are there to mark the answers? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM A multiple-choice test has 6 questions, each with 4 possible Are the objects answers. How many ways are there to mark the answers? Yes you’re counting all the same type? Can you repeat No a selection? Yes No Are you putting the objects in Use order? Fundamental Counting Principle Yes No w x y z ways ways ways ways Are you Use arranging ALL Combinations: nCr the objects? = w·x·y·z ways Yes No 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 4 6 = 4,096 Use Use Factorial: n! Permutations: nPr
32 nd International Conference on Technology in Collegiate Mathematics ORLANDO, FL | MARCH 12-14 #ICTCM In how many ways can you line up 6 people for a photograph?
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