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Counting Theory (Permutation and Combination)

Counting Theory (Permutation and Combination). Starter 6.0.1. Suppose you have 6 different textbooks in your backpack that you want to put on a bookshelf. How many ways can the 6 books be arranged on the shelf?. Objectives.

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Counting Theory (Permutation and Combination)

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  1. Counting Theory(Permutation and Combination)

  2. Starter 6.0.1 • Suppose you have 6 different textbooks in your backpack that you want to put on a bookshelf. How many ways can the 6 books be arranged on the shelf?

  3. Objectives • Use organized lists and tree (branching) diagrams to list all possible outcomes of a trial. • Identify whether permutation or combination is appropriate to count the number of outcomes of a trial. • Use formulas or calculator commands to evaluate permutation and combination problems.

  4. Counting Outcomes • There are three principle ways to count all the outcomes of a trial. • Draw a tree diagram • Often a simple multiplication is enough • Systematically write all possibilities • Use permutation and combination techniques

  5. Example: Tossing Coins • Three coins are tossed (or one coin is tossed three times) and the outcome of heads or tails is observed. • Draw a tree diagram (also called a branching diagram) that shows all possible outcomes. • State a conclusion: How many equally likely outcomes are there in this problem?

  6. Tree Diagram for 3 Coins First Toss Second Toss Third Toss H T H H T H T H H T T T H • So there are 8 different equally likely outcomes. T

  7. Write an Organized List • For the coin-toss problem we just did, write an organized list that shows all possible outcomes (like HHH etc) • Here is one possible organization • HHH, HHT, HTH, HTT • TTT, THT, TTH, THH • There are other ways to organize • Any method that is systematic (so that no outcomes are missed) can work

  8. Permutations of n objects • Return to the bookshelf question. • Suppose we change the problem to arranging 10 books on the shelf. Now how many arrangements are there? • 10 x 9 x 8 x … = 3,628,800 • The shorthand notation for this is 10! (factorial) • In general, there are n! ways to arrange n objects • This is called the permutation of n objects • The key idea is that order matters

  9. Arranging fewer than all the objects • What if there are only 4 slots available on the bookshelf for the 10 books? • Then there are 10 x 9 x 8 x 7 = 5040 ways to arrange 4 books out of a group of 10 • Notice that this could be viewed as • If we let n = the total number of objects and r = the number chosen and arranged, then we could conclude that the number of ways to arrange n objects taken r at a time is • This can be quickly evaluated by (nPr) • Try it now on your calculator with n = 10 and r = 4

  10. Example • How many three-letter “words” can be made from the letters A, B, C, and D? • You can use your calculator to answer this. • What are n and r in this problem? • Don’t worry that many of them are not real words; we don’t care in this context. • Write an organized list of all the possible “words” • Be systematic; be sure you write them all.

  11. Three-letter “words” • Notice that being organized helps find all 24 permutations • Notice also that ABC is different from ACB because in permutation order matters • Suppose we don’t care about order. Then we are looking at combination, not permutation. • How many combinations of three letters can be made from an alphabet with four letters?

  12. Three-letter “words” • When order does not matter, ABC is the same as ACB (etc.), so there are only 4 combinations in the 24 permutations. • They can be seen in the 4 columns • Notice that there are 3! (which is r!) permutations of each combination. • They can be seen in the 6 rows • So to get the number of combinations of n objects taken r at a time, divide permutations by r! • The formula is • The calculator command is (nCr) Try it now.

  13. Examples • How many ways are there to form a 3 member subcommittee from a group of 12 people? • How many ways are there to choose a president, vice-president, and secretary from a group of 12 people?

  14. More Examples • There are 5 cabins in the woods at a certain vacation spot. Each cabin has a path that leads to each of the other cabins. How many paths are there in all? • This is the combination of 5 things taken 2 at a time (order does not matter), so 5C2=10 • There are 100 communications satellites orbiting earth. Each satellite needs a transmit and receive channel to talk to each of the other satellites. How many channels are needed? • This time AB is different from BA, so use permutation: 100P2=9900 • How many pentagons can be drawn from the vertices of a regular 13-gon? • Combination: 13C5=1287

  15. Objectives • Use organized lists and tree (branching) diagrams to list all possible outcomes of a trial. • Identify whether permutation or combination is appropriate to count the number of outcomes of a trial. • Use formulas or calculator commands to evaluate permutation and combination problems.

  16. Homework • Create and solve two story problems which illustrate the differences between combinations and permutations. • Create and solve a problem involving the permutation of n things taken r at a time. • Create and solve a problem involving the combination of n things taken r at a time.

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