SECTION 13.2 Permutations and Combinations 917 1 Solve Counting Problems Using Permutations Involving n Distinct Objects DEFINITION Permutation A permutation is an ordered arrangement of r objects chosen from n objects. THEOREM Permutations: Distinct Objects with Repetition The number of ordered arrangements of r objects chosen from n objects, in which the n objects are distinct and repetition is allowed, is n .r Three types of permutations are discussed: • The n objects are distinct (different), and repetition is allowed in the selection of r of them. [Distinct, with repetition] • The n objects are distinct (different), and repetition is not allowed in the selection of r of them, where ≤ r n. [Distinct, without repetition] • The n objects are not distinct, and all of them are used in the arrangement. [Not distinct] We take up the first two types here and deal with the third type at the end of this section. The first type of permutation ( n distinct objects, repetition allowed) is handled using the Multiplication Principle. Counting Airport Codes [Permutation: Distinct, with Repetition] The International Airline Transportation Association (IATA) assigns three-letter codes to represent airport locations. For example, the airport code for Ft. Lauderdale, Florida, is FLL. Notice that repetition is allowed in forming this code. How many airport codes are possible? Solution EXAMPLE 1 An airport code is formed by choosing 3 letters from 26 letters and arranging them in order. In the ordered arrangement, a letter may be repeated. This is an example of a permutation with repetition in which 3 objects are chosen from 26 distinct objects. The task of counting the number of such arrangements consists of making three selections. Each selection requires choosing a letter of the alphabet (26 choices). By the Multiplication Principle, there are ⋅ ⋅ = = 26 26 26 26 17,576 3 possible airport codes. The solution given to Example 1 can be generalized. Now Work PROBLEM 33 Now let’s consider permutations in which the objects are distinct and repetition is not allowed. Forming Codes [Permutation: Distinct, without Repetition] Suppose that a three-letter code is to be formed using any of the 26 uppercase letters of the alphabet, but no letter is to be used more than once. How many different threeletter codes are there? EXAMPLE 2 (continued)
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