Additional Topics in Probability and Counting 3.4 168 CHAPTER 3 Probability Permutations Combinations Applications of Counting Principles What You Should Learn How to find the number of ways a group of objects can be arranged in order How to find the number of ways to choose several objects from a group without regard to order How to use counting principles to find probabilities Sudoku Number Puzzle 6 8 2 5 5 5 2 2 4 4 2 2 8 8 8 7 1 9 9 9 9 1 1 1 1 1 3 3 7 7 7 7 6 6 6 6 Permutations In Section 3.1, you learned that the Fundamental Counting Principle is used to find the number of ways two or more events can occur in sequence. An application of the Fundamental Counting Principle is finding the number of ways that n objects can be arranged in order. An ordering of n objects is called a permutation. A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!. DEFINITION The expression n! is read as n factorial. If n is a positive integer, then n! is defined as follows. n! = n# 1n - 12 # 1n - 22 # 1n - 32g3# 2# 1 As a special case, 0! = 1. Here are several other values of n!. 1! = 1 2! = 2# 1 = 2 3! = 3# 2# 1 = 6 4! = 4# 3# 2# 1 = 24 Finding the Number of Permutations of n Objects The objective of a 9 * 9 Sudoku number puzzle is to fill the grid so that each row, each column, and each 3 * 3 grid contain the digits 1 through 9 without repetition. How many different ways can the first row of a blank 9 * 9 Sudoku grid be filled? SOLUTION The number of permutations is 9! = 9# 8# 7# 6# 5# 4# 3# 2# 1 = 362,880. So, there are 362,880 different ways the first row can be filled. TRY IT YOURSELF 1 The Big 12 is a collegiate athletic conference with 10 schools: Baylor, Iowa State, Kansas, Kansas State, Oklahoma, Oklahoma State, Texas, Texas Christian, Texas Tech, and West Virginia. How many different final standings are possible for the Big 12’s football teams? Answer: Page A38 You may want to choose some of the objects in a group and put them in order. Such an ordering is called a permutation of n objects taken r at a time. The number of permutations of n distinct objects taken r at a time is nPr = n! 1n - r2! , where r … n. Permutations of n Objects Taken r at a Time EXAMPLE 1 Study Tip Notice that small values of n can produce very large values of n!. For instance, 10! = 3,628,800. Be sure you know how to use the factorial key on your calculator. For help with factorials, see Integrated Review at MyLab Statistics
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