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    Title Page
    Contents at a Glance
    Contents
    About the Authors
    About the Technical Reviewer
    Introduction
    CHAPTER 1: Overview of PHP Data Types and Concepts
    CHAPTER 2: Overview of Classes, Objects, and Interfaces
    CHAPTER 3: Performing Math Operations
    CHAPTER 4: Working with Arrays
    4-1. Creating Arrays
    4-2. Accessing Array Elements
    4-3. Creating Multidimensional Arrays
    4-4. Using Array Keys
    4-5. Initializing an Array As a Range or Sequence of Values
    Outputting Arrays
    4-6. Outputting an Array As a String
    4-7. Outputting Using array_values() and array_keys() for Backward Compatibility
    4-8. Outputting an Array As a Tree
    Adding New Elements to Arrays
    4-9. Adding an Element to the End of an Array
    4-10. Appending One Array to Another
    4-11. Comparing Arrays
    4-12. Adding an Element to the Beginning of an Array
    4-13. Inserting New Values at an Arbitrary Point in an Indexed Array
    Getting and Setting the Size of an Array
    4-14. Counting Array Elements
    4-15. Setting an Array's Size
    Traversing Arrays
    4-16. Looping Through an Associative Array Using foreach
    4-17. Looping Through a Compact Indexed Array Using for and count()
    4-18. Looping Through a Sparse Array
    Removing Elements from Arrays
    4-19. Removing the First or Last Element from an Array
    4-20. Removing One or More Arbitrary Array Elements
    4-21. Extracting a Portion of an Array
    4-22. Extracting Values from Arrays with extract()
    4-23. Extracting Values from an Array Using list()
    4-24. Combining Arrays
    4-25. Obtaining Array Keys and Values
    4-26. Working with Unique Values
    4-27. Getting and Displaying Counts of Array Values
    Finding and Working with Array Values
    4-28. Determining Whether an Element Is in an Array
    4-29. Testing for the Existence of a Key in an Array
    4-30. Obtaining Array Keys with a Given Value
    4-31. Finding the Greatest and Least Values in an Array
    4-32. Finding the Sum and Average of the Values in an Array
    Applying Functions to Arrays
    4-33. Applying Functions to Array Elements Using array_walk()
    4-34. Applying Functions to Array Elements Using array_map()
    4-35. Filtering Arrays Using array_filter()
    Sorting Arrays
    4-36. Sorting an Array by Its Values
    4-37. Sorting an Array by Its Keys
    4-38. Reversing an Array Using arsort()
    4-39. Reversing an Array Using krsort()
    4-40. Reversing an Array Using array_reverse()
    4-41. Randomizing an Array Using shuffle(), kshuffle(), and array_rand()
    4-42. Sorting an Array Using Comparison Functions
    4-43. Sorting Multidimensional Arrays
    4-44. Sorting Multiple Arrays
    Finding Permutations and Combinations
    4-45. Finding All Permutations of an Array's Elements
    4-46. Finding All Combinations of an Array's Elements
    Summary
    Looking Ahead
    CHAPTER 5: Working with Dates and Times
    CHAPTER 6: Working with Strings
    CHAPTER 7: Working with Files and Directories
    CHAPTER 8: Working with Dynamic Imaging
    CHAPTER 9: Using Regular Expressions
    CHAPTER 10: Working with Variables
    CHAPTER 11: Using Functions
    CHAPTER 12: Understanding Web Basics
    CHAPTER 13: Creating and Using Forms
    CHAPTER 14: Working with Markup
    CHAPTER 15: Using MySQL Databases in PHP 5
    CHAPTER 16: Communicating with Internet Services
    Index
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    CHAPTER 4: Working with Arrays > Finding Permutations and Combinations

    Finding Permutations and Combinations

    In order to understand recursion, one first must understand recursion.

    —Anonymous

    Both combinations and permutations of a set of values involve different arrangements of those values. In this section, we will define what these terms mean and show you how to do the following:

    • Calculate numbers of possible permutations and combinations of an array's elements
    • Generate permutations and combinations


      

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