1089 11.6 Basics of Counting Theory In using the fundamental principle of counting, products such as 5 # 4 # 3 # 2 # 1 occur often. We use the symbol n! (read “n-factorial ”), for any counting number n, as follows. n! =n1n −12 1n −22 P 132 122 112 Examples: 5 # 4 # 3 # 2 # 1 = 5! and 3 # 2 # 1 = 3! By the definition of n!, n31n - 12!4 = n! for all natural numbers n Ú 2. It is convenient to have this relation hold also for n = 1, and so, by definition, 0! =1. EXAMPLE 3 Arranging r of n Items 1r *n2 Suppose the teacher in Example 2 wishes to place only 3 of the 5 books in a row. How many arrangements of 3 books are possible? SOLUTION The teacher still has 5 ways to fill the first spot, 4 ways to fill the second spot, and 3 ways to fill the third. Only 3 books will be used, so there are only 3 spots to be filled (3 events) instead of 5. Again, we use the fundamental principle of counting. 5 # 4 # 3 = 60 arrangements S Now Try Exercise 13. Permutations Because each ordering of three books is considered a different arrangement, the number 60 in the preceding example is called the number of permutations of 5 things taken 3 at a time, written P15, 32 = 60. A permutation of n elements taken r at a time is one of the arrangements of r elements from a set of n elements. Generalizing, the number of permutations of n elements taken r at a time, denoted by P1n, r2, is given as follows. P1n, r2 = n1n - 12 1n - 22 g 1n - r + 12 P1n, r2 = n1n - 12 1n - 22 g 1n - r + 12 1n - r2 1n - r - 12 g 122 112 1n - r2 1n - r - 12 g 122 112 P1n, r2 = n! 1n - r2! Permutations of n Elements Taken r at aTime If P1n, r2 denotes the number of permutations of n elements taken r at a time, with r … n, then the following holds true. P1n, r2 = n! 1 n −r2! An alternative notation for P1n, r2 is nPr .
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