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3.7 Counting Techniques: Permutations

3.7 Counting Techniques: Permutations. If we line up all the students in this class, what is the probability that Aninidita and Hanine are standing together? If I randomly create groups of 3, what is the probability that Marian, Ngoc, and Sumi are all in the same group?

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3.7 Counting Techniques: Permutations

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  1. 3.7 Counting Techniques: Permutations

  2. If we line up all the students in this class, what is the probability that Aninidita and Hanine are standing together? • If I randomly create groups of 3, what is the probability that Marian, Ngoc, and Sumi are all in the same group? all in different groups? • These require techniques to count the outcomes of more complex experiments • combinatorics: permutations and combinations

  3. How many different ways can we line up the students in this class? Let’s try it! (or at least, let’s try a small sample…)

  4. B C A C B A B C A C B A C B A Counting Techniques • Example: Placing 3 objects, A, B, C in line for a game Resulting Order First Second Third A BC ACB BAC BCA CAB CBA # different orders possible = Multiplicative = 3 x 2 x 1 = 3! 6

  5. Factorial Notation n! = n(n-1)(n-2)…(3)(2)(1) 4! = 4(3)(2)(1) = 24 • n! represents the number of ways n different objects can be ordered • that is, the number of ways n objects can be selected to create ordered arrangements of size n

  6. Example 1 • How many ways can we arrange Andrey, Bibi, Charles, Deb, and Eunice in a line-up? Note: you can use the x! button on your calculator!

  7. Permutation • A ordered arrangement of n different objects taken r at a time (n! means we arrange all n objects; a permutation only arranges r of those n objects) • The total number of possible arrangements, or permutations, of r objects taken from a set of size n is denoted by P(n, r) = nPr

  8. How? • How many ways can we arrange 3 objects from a group of 6? 6 5 4 We write:

  9. Example 3 • How many ways can we arrange 4 books out of 10?

  10. Error on the calculator! Example 4 You can also use the nPr button on your calculator. • How many ways can we arrange 30 objects out of 100? nPr Type: 100 30 =

  11. Permutations with Identical Elements • Consider DOL1L2 • # permutations = 4! = 24 • Examples: DOL1L2DOL2L1 DL1L2O DL2L1O L1L2OD L2L1OD DOLLDOLL DLLO DLLO LLOD LLOD • The L’s are the same • Out of 24 arrangements, only 12 are different from each other • The number of different arrangements is

  12. Permutations with Identical Elements • The number of permutations of a set of n objects containing a identical objects of one kind, b identical objects of a second kind, c identical objects of a third kind, etc., is

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