918 CHAPTER 13 Counting and Probability Solution Some of the possibilities are ABC, ABD, ABZ, ACB, CBA, and so on. The task consists of making three selections. The first selection requires choosing from 26 letters. Since no letter can be used more than once, the second selection requires choosing from 25 letters. The third selection requires choosing from 24 letters. (Do you see why?) By the Multiplication Principle, there are ⋅ ⋅ = 26 25 24 15,600 different three-letter codes with no letter repeated. For the second type of permutation, we introduce the following notation. The notation P n r , ( ) represents the number of ordered arrangements of r objects chosen from n distinct objects, where ≤ r n and repetition is not allowed. For example, the question posed in Example 2 asks for the number of ways in which the 26 letters of the alphabet can be arranged, in order, using three nonrepeated letters. The answer is ( ) = ⋅ ⋅ = P 26, 3 26 25 24 15,600 1st 2nd 3rd r th P n r n n n n r n n n n r , 1 2 1 1 2 1 ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) = ⋅ − ⋅ − ⋅…⋅ − − = ⋅ − ⋅ − ⋅…⋅ − + Lining People Up In how many ways can 5 people be lined up? Solution EXAMPLE 3 The 5 people are distinct. Once a person is in line, that person will not be repeated elsewhere in the line; and, in lining people up, order is important.This is a permutation of 5 objects taken 5 at a time, so 5 people can be lined up in ( ) = ⋅ ⋅ ⋅ ⋅ = P 5,5 5 4 3 2 1 120ways 5 factors Now Work PROBLEM 35 To arrive at a formula for ( ) P n r , , note that the task of obtaining an ordered arrangement of n objects in which only ≤ r n of them are used, without repeating any of them, requires making r selections. For the first selection, there are n choices; for the second selection, there are −n 1 choices; for the third selection, there are −n 2 choices; . . . ; for the rth selection, there are ( ) − − n r 1 choices. By the Multiplication Principle, this means RECALL n n n 0! 1, 1! 1, 2! 2 1, , ! 1 3 2 1 ( ) = = = ⋅ … = − ⋅…⋅ ⋅ ⋅ j This formula for ( ) P n r , can be compactly written using factorial notation. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ⋅ − ⋅ − ⋅…⋅ − + = ⋅ − ⋅ − ⋅…⋅ − + ⋅ − ⋅…⋅ ⋅ ⋅ − ⋅…⋅ ⋅ ⋅ = − P n r n n n n r n n n n r n r n r n n r , 1 2 1 1 2 1 3 2 1 3 2 1 ! !
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