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HOMEWORK RULES Lots of confusion Submit via email by noon on Saturday Please learn how to use a scanner and scan your homework in. I need procedures not only the answers. Thank you! LAST LECTURE Basic Counting Principle Arithmetic Sequences Geometric Sequences Shifted Sequences 2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049 PERMUTATION WB Math Group TOOLS Trees Simple Algebra ROCK PAPER SCISSORS Player2 Player1 Rock Rock Paper Scissors Rock Paper Paper Scissors Rock Scissors Paper Scissors ROCK PAPER SCISSORS LIZARD SPOCK ROCK PAPER SCISSORS LIZARD SPOCK ROCK PAPER SCISSOR ON STEROIDS 1.3 trillion permutation A SIMPLE PROBLEM You have three shirts and four pairs of pants. How many outfits consisting of one shirt and one pair of pants can you make? Shirts – S1, S2, S3, Pants – P1, P2, P3, P4 S1 P1 P2 S2 P3 P4 3 P1 P2 4 S3 P3 P4 12 P1 P2 P3 P4 ANOTHER SIMPLE PROBLEM How many three letter words can be formed by letters in DOG? D O G O G G O D G G D D O O D 3 2 1 6 MULTIPLICATION PRINCIPAL - number of ways A and B can occur - number of ways A can occur - number of ways B can occur Event A Event B 1 1 2 … 2 1 2 … … … 1 2 … SOME SIMPLE PROBLEMS In how many ways can we arrange four different books on a shelf? Your math club has 20 members. In how many ways can it select a president, a vice-president and a treasurer if no member can hold more than one office? Previous problem without the limitation of no member can hold more than one office? In how many ways can we arrange n different books, where n is a positive integer? A LITTLE ALGEBRA Factorial: Compute the following: 9!/8! 42!/40! 8! – 7! (hint: law of distribution) For each of 8 colors, you have one shirt and one tie of that color. How many shirt-and-tie outfits can you make if you refuse to wear a shirt and a tie of the same color? Can you express your answer in terms of factorials? A SLIGHTLY HARDER PROBLEM A club has n members, where n is a positive integer. We want to choose r different officers of the club, where r < n, such that no member can hold more than one office. In how many ways can we fill the first office? Once we’ve filled the first office, in how many ways can we fill the second office? If we proceed as in the previous two steps, in how many ways can we fill the office? In how many ways can we choose the r officers? PERMUTATION The number of permutations of size r from a group of n objects is BRAIN EXERCISE Compute the following P(8, 3) P(20, 4) P(30, 1) P(6,5) P(50, 3) What is P(n, n) for any positive integer n? does it make sense? What is 0! Simplify SOME EXERCISES You have 5 shirts, 6 pairs of pants and hats, how many outfits can you make consisting of one shirt, one pair of pants and one hat? How many 3-letter combinations can be formed if the second letter must be a vowel (a,e,i,o,u) and the third letter must be different from the first letter? In how many ways can you order 7 different colored hats in a row? Our basketball team has 12 members, each of whom can play any position. In how many ways can we choose a starting lineup consisting of a center, a power forward, a shooting forward, a point guard, and a shooting guard? SOME HARDER PROBLEMS How many permutations does GAUSS have? How may different license plates can be made where each plate has three distinct lower case English letters? How many integers between 100 and 999 inclusive consist of distinct odd digits? How many integers between 100 and 999 have distinct digits? There are 10 points where no three points are on one line. How many straight lines can be made by connecting two points? 10 boys and 9 girls sit in a row of 19 seats. How many ways can this be done if All boys sit next to each other and all girls sit next to each other Each child has only neighbors of the opposite gender? 10 boys and 9 girls sit in a row of 19 seats. How many ways can this be done if Each child has only neighbors of the opposite gender?