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Combinatorics

Combinatorics. CSLU 1100.003 Fall 2007 Cameron McInally cameron.mcinally@nyu.edu Fordham University. Combinatorics. Counting Figuring out how many. This can be a bit confusing. The best way to learn it? Practice problems!. Combinatorics. Common Sense

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Combinatorics

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  1. Combinatorics CSLU 1100.003 Fall 2007 Cameron McInally cameron.mcinally@nyu.edu Fordham University

  2. Combinatorics • Counting • Figuring out how many. • This can be a bit confusing. • The best way to learn it? Practice problems!

  3. Combinatorics • Common Sense • “If there are 10 boys and 12 girls in a class, how many people are there altogether?” 10 + 12 = 22

  4. Combinatorics • Common Sense • “If a car dealership sells 3 different models of cars and offers them in 4 different colors, how many different ways can you purchase a car?” 3 * 4 = 12

  5. Combinatorics • A little more difficult • “If a New York State license plate consists of 3 letters followed by 4 numbers, how many different license plate possibilities are there?” 26 * 26 * 26 * 10 * 10 * 10 * 10 = 263 * 104 = 175760000

  6. Combinatorics • Counting • When counting possible outcomes, we may wish to combine terms. We can combine terms using And or Or.

  7. Combinatorics • And • If terms combine with “AND” then you multiply the numbers. • Example: Pick one letter and one number between 0 and 9. How many possible combinations are there?

  8. Combinatorics • Or • If terms combine with “OR” then you add the numbers. • Example: Pick one letter or one number between 0 and 9. How many possible combinations are there?

  9. Combinatorics • The counting process considers the number of different ways you can select items from a group of items. • There are 4 types of groups we can select from: (Note: Memorize the definition of each and your life will be much easier!!!) • Unordered List • Ordered List • Set • Permutation

  10. Combinatorics • Permutation • An ordered list of elements. We CANNOThave duplicates. We DO care about the order of elements in a set. • If the list is of size n and we want to know how many ways we can select r elements…

  11. Combinatorics • Permutation Example • We have 6 unique Xbox games. We let 6 friends F={f0,f1,f2,f3,f4,f5} each borrow 1 game. How many different ways can we distribute the 6 games to 6 friends?

  12. Combinatorics • Ordered List • An ordered list of elements. We CANhave duplicates. We DO care about the order of elements in a set. • If the list is of size n and we want to know how many ways we can select r elements…

  13. Combinatorics • Ordered List Example • How many numbers are there between 0 and 999, inclusive? That is to say, how many permutations of three digits exist, if each digit is between 0 and 9?

  14. Combinatorics • Set • It’s a set. We CANNOThave duplicates. We DO NOT care about the order of elements in a set. • If the list is of size n and we want to know how many ways we can select r elements…

  15. Combinatorics • Set Example • We have 9 friends on MySpace. We want to fill our top 8 spots. How many different combinations of 8 friends can we picks?

  16. Combinatorics • Unordered List • Kind of like a set. We CANhave duplicates. We DO NOT care about the order of elements in a set. • If the list is of size n and we want to know how many ways we can select r elements…

  17. Combinatorics • Unordered List Example • There are 5 different colors of iPod nanos. We want to buy two. How many different combinations of colors could we pick?

  18. Combinatorics • A quick reference Does Order Matter? Are Repetitions Allowed?

  19. Combinatorics • Permutation Practice Problem • There are 3 horses running in a race. What are the possible outcomes of the horse race. {123,132,213,231,312,321} • So, there are 6 possible outcomes and…

  20. Combinatorics • Ordered List Practice Problem • We have 3 bits. Each bit can have either the value 0 or 1. How many unique 3 digit strings can we have? {000,001,010,011,100,101,110,111} • So, there are 8 possible outcomes and…

  21. Combinatorics • Set Practice Problem • We have 3 lamps and 2 electric sockets. How many different ways can we light 2 lamps? {011,101,110} • So, there are 3 possible outcomes and…

  22. Combinatorics • Unordered Practice Problem • There are three different brands of beer at a bar. You want to buy 2 beers. How many ways can you buy 2 beers? {ab,ac,bc,aa,bb,cc} • So, there are 6 possible outcomes and…

  23. Homework(Always Due in One Week) Combinatorics • Read Section 5.1, 5.2. Skim Section 5.3. • Complete Section 5.1 pages 382 : 4 (a - e) • Complete Section 5.2 pages 395-396 : 1 (a,b), 3 (a,b), 9 (a,b) • Solve the “Towers of Hanoi” problem for n = 1 through 4, page 420. Write out each step! Approximately how many moves would it take to solve this for n = 64?

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