Permutations and Combinations Additional Problems . The slides for this text are organized into chapters. This lecture covers Chapter 1. Chapter 1: Introduction to Database Systems Chapter 2: The Entity-Relationship Model Chapter 3: The Relational Model Chapter 4 (Part A): Relational Algebra Chapter 4 (Part B): Relational Calculus Chapter 5: SQL: Queries, Programming, Triggers Chapter 6: Query-by-Example (QBE) Chapter 7: Storing Data: Disks and Files Chapter 8: File Organizations and Indexing Chapter 9: Tree-Structured Indexing Chapter 10: Hash-Based Indexing Chapter 11: External Sorting Chapter 12 (Part A): Evaluation of Relational Operators Chapter 12 (Part B): Evaluation of Relational Operators: Other Techniques Chapter 13: Introduction to Query Optimization Chapter 14: A Typical Relational Optimizer Chapter 15: Schema Refinement and Normal Forms Chapter 16 (Part A): Physical Database Design Chapter 16 (Part B): Database Tuning Chapter 17: Security Chapter 18: Transaction Management Overview Chapter 19: Concurrency Control Chapter 20: Crash Recovery Chapter 21: Parallel and Distributed Databases Chapter 22: Internet Databases Chapter 23: Decision Support Chapter 24: Data Mining Chapter 25: Object-Database Systems Chapter 26: Spatial Data Management Chapter 27: Deductive Databases Chapter 28: Additional Topics

Permutations and Combinations with and without Repetition Discrete Mathematics and its Applications 9/8/2018 Permutations and Combinations with and without Repetition (c)2001-2002, Michael P. Frank

n is number of distinct classes of objects in the original bag! - r-permutation without repetition - order matters (r distinguishable slots) - without replacement (n distinguishable objects) - r-combination without repetition - order does not matter (r indistinguishable slots) - without replacement (n distinguishable objects) n! / r! (n-r)! n! / (n-r)! - r-permutation with repetition - order matters (r distinguishable slots) - with replacement (n distinct classes of indistinguishable objects) n^r - r-combination with repetition - order does not matter (r indistinguishable slots) - with replacement (n distinct classes of indistinguishable objects) (n+r-1)! / r! (n-1)!

Combinations or Permutations? 1. In how many ways can you choose 5 out of 10 friends to invite to a dinner party? Solution: Does the order of selection matter? NO If you choose friends in the order A,B,C,D,E or A,C,B,D,E the same set of 5 was chosen, so we conclude that the order of selection does not matter. We will use the formula for combinations since we are concerned with how many subsets of size 5 we can select from a set of 10. C(10,5) =

Permutations or Combinations? How many ways can you arrange 10 books on a bookshelf that has space for only 5 books? Does order matter? The answer is yes since the arrangement ABCDE is a different arrangement of books than BACDE. We will use the formula for permutations. We need to determine the number of arrangements of 10 objects taken 5 at a time so we have P(10,5) = 10(9)(8)(7)(6)=30,240

Lottery problem A certain state lottery consists of selecting a set of 6 numbers randomly from a set of 49 numbers. To win the lottery, you must select the correct set of six numbers. How many possible lottery tickets are there? Solution. The order of the numbers is not important here as long as you have the correct set of six numbers. To determine the total number of lottery tickets, we will use the formula for combinations and find C(49, 6), the number of combinations of 49 items taken 6 at a time. C(49,6) = 13,983,816

Examples How many ways can a 3-person subcommittee be selected from a committee of a seven people? The number of ways that a three-person subcommittee can be selected from a seven member committee is the number of combinations (since order is not important in selecting a subcommittee) of 7 objects 3 at a time. This is:

Example (cont) How many ways can a president, vice-president, and secretary can be chosen from a committee of 7 people? The number of ways a president, vice-president, and secretary can be chosen from a committee of 7 people is the number of permutations (since order is important in choosing 3 people for the positions) of 7 objects 3 at a time. This is: P(7,3)

The five spades can be selected in C13,5 ways and the two Problem From a standard 52-card deck, how many 7-card hands have exactly 5 spades and 3 hearts? The five spades can be selected in C13,5 ways and the two hearts can be selected in C13,2 ways. Applying the Multiplication Principle, we have: Total number of hands

Order doesn’t matter and Repetitions are allowed Problem How many ways are there to select five bills from a cash box containing $1, $2, $5, $10, $20, $50 and $100 bills, such that the bills are indistinguishable and the order in which they are selected is unimportant. (there are also at least 5 bills of each kind). Solution Order doesn’t matter and Repetitions are allowed This is like drawing colored balls with replacement. The colors correspond to the values. Since the order doesn’t matter we have: C(7+5-1,5)=462

Combinations with Repetition Discrete Mathematics and its Applications 9/8/2018 Combinations with Repetition Approach: Place five markers in the compartments i.e., # ways to arrange five stars and six bars ... Solution: Select the positions of the 5 stars from 11 possible positions ! C(n+r-1,5)= C(7+5-1,5)=C(11,5) n=7 r=5 compartments and dividers markers (c)2001-2002, Michael P. Frank

exercise A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?

exercise A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Center: Forwards: Guards: Thus, the number of ways to select the starting line up is 2*10*6 = 120.

Problem A cookie shop has 4 kinds of cookies and we want to pick 6. In how many ways we can pick them? We don’t care about the order and cookies from one kind are indistinguishable. This is precisely the problem we saw to solve the r-combination with repetition C(6+4-1,6)=84.

Problem In how many ways can we place 10 indistinguishable items in 8 distinguishable boxes. This is precisely the problem we saw to solve the r-combination with repetition: C(10+8-1,10)

How many different “words” can we create by reordering SUCCESS ? Problem How many different “words” can we create by reordering SUCCESS ? Total number of permutations is 7!. However permuting the 3 S’s does not create a new word, idem 2 C’s: 7!/3! 2!

a) How many ways if the boxes are numbered? Problem Math teacher has 40 issues of a journal and packs them into 4 boxes,10 issues each. a) How many ways if the boxes are numbered?  assign boxes to issues: 40! / (10!)^4 b) How many ways if the boxes are indistinguishable.  There are 4! ways to label the boxes, once we have distributed them in unlabelled boxes. Since the number of ways to distribute them in labeled boxes is given by a) i.e..40! / (10!)^4 we get 40! / (10!)^4 4!.

How many different cross terms will we generate when we multiply out: Problem How many different cross terms will we generate when we multiply out: (x1+x2+...+xm)^n ? How many different exponents are there of the sort x1^n1 x2^n2 ... xm^nm with n1+n2+...+nm=n. Equivalent to : how many different way are there to put n balls in m boxes: C(n+m-1,n)

Tie-up 3 boys as one , thus it is Problem In how many ways can 7 girls and 3 boys line up, if the boys must stand next to each other? Tie-up 3 boys as one , thus it is about ordering 8 “things” and then choosing an order for the boys inside the one “big thing”. 8!*3!

Problem Suppose an operating system has a queue of 3 low priority and 5 high priority processes ready to run. In how many ways could these processes be ordered for execution if 2 low priority processes are not allowed to be executed back to back? Order H in 5! Ways To get HHHHH Then _H_H_H_H_H_ Order L in 3! Ways 6p3 for the 3 spots where we put the Ls. 5!*3!* 6p3