Combinatorics CSLU Fall 2007 Cameron McInally 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 – “If there are 10 boys and 12 girls in a class, how many people are there altogether?” = 22

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

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 = 26 3 * 10 4 =

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

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?

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?

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

Combinatorics Permutation –An ordered list of elements. We CANNOT have 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…

Combinatorics Permutation Example –We have 6 unique Xbox games. We let 6 friends F={f 0,f 1,f 2,f 3,f 4,f 5 } each borrow 1 game. How many different ways can we distribute the 6 games to 6 friends?

Combinatorics Ordered List –An ordered list of elements. We CAN have 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…

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?

Combinatorics Set –It’s a set. We CANNOT have 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…

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?

Combinatorics Unordered List –Kind of like a set. We CAN have 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…

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?

Combinatorics A quick reference YesNo YesOrdered List Unordered List NoPermutationSet Does Order Matter? Are Repetitions Allowed?

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…

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…

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…

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…

Always Due in One Week Homework (Always Due in One Week) Read Section 5.1, 5.2. Skim Section 5.3. Complete Section 5.1 pages 382 : 4 (a - e) Complete Section 5.2 pages : 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? Combinatorics