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Unformatted text preview: Ch2/MATH0211/YMC/2009-10 1 Chapter 2. Permutations and Combinations 2.1. The Basic Principle of Counting In this section we introduce the basic principle of counting. Let us first consider the following example. Example 2.1 Suppose there are two roads connecting cities A and B , three connecting B and C , and five connecting C and D . How many different routes are possible if we try to drive from A , to B to C and then to D ? Solution . Starting from city A , we can choose any of the two routes, then from B we can choose any of the three routes, and from C we can choose any of the five routes. Multiplying them together gives the total, that is, 2 × 3 × 5 = 30 different routes. 2 This multiplication procedure can be generalized into the following basic principle of counting: The Basic Principle of Counting Suppose that a procedure involves a sequence of k stages. Let n 1 be the number of ways the first can occur and n 2 be the number of ways the second can occur. Continuing in this way, let n k be the number of ways the k th stage can occur. Then the total number of different ways the procedure can occur is n 1 · n 2 ··· n k . Example 2.2 Suppose a coin is tossed twice and then a die is rolled. How many different results can occur? Solution . Here we have a three-stage procedure – tossing a coin twice and then rolling a die. Each of the first two stages (the coin toss) has two possible outcomes. The third stage (rolling a die) has six possible outcomes. By the basic principle of counting, the number of different results of the procedure is 2 × 2 × 6 = 24 . 2 Ch2/MATH0211/YMC/2009-10 2 Example 2.3 How many numbers of three different digits each can be formed by choosing the digits 1 , 2 , 3 , 4 , 5 and 6 ? Solution . To form a three-digit number, we must fill the positions- - - with different digits. For the first position we have six choices. After filling that position with some digit, we can fill the second position with any of the remaining five digits. Finally there are four choices for the last position. By the basic principle of counting, the total number of three different digits that can be formed by choosing from the six given digits is 6 × 5 × 4 = 120 . 2 Remark If repetitions are allowed, the total number would then be 6 × 6 × 6 = 216 . 2.2. Permutations Definition 2.4 An ordered selection of r objects, without repetition, taken from n distinct objects is called a permutation of r objects chosen from a set of n objects . The number of such permutations is denoted by n P r ....
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