College Algebra & Trigonometry, 2018a

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7.1. THE FUNDAMENTAL PRINCIPLE OF COUNTING 311 10) A combination lock is numbered from 0 to 30. Each combination consists of three numbers in succession. Successive numbers must be different, but the first and third can be the same. How many different combinations are possible?
312 CHAPTER 7. COMBINATORICS 7.2 Factorial Notation and Permutations In considering the number of possibilities of various events, particular scenarios typically emerge in different problems. One of these scenarios is the multiplication of consecutive whole numbers. For example, given the question of how many ways there are to seat a given number of people in a row of chairs, there will obviously not be repetition of the individuals. So, if we wanted to know how many different ways there are to seat 5 people in a row of five chairs, there would be 5 choices for the first seat, 4 choices for the second seat, 3 choices for the third seat and so on. 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120 choices In these situations the 1 is sometimes omitted because it doesn’t change the value of the answer. This process of multiplying consecutive decreasing whole numbers is called a “factorial.” The notation for a factorial is an exclamation point. So the problem above could be answered: 5! = 120. By definition, 0! = 1. Although this may not seem logical intuitively, the definition is based on its application in permutation problems. A“permutation” uses factorials for solving situations in which not all of the possibilities will be selected. So, for example, if we wanted to know how many ways can first, second and third place finishes occur in a race with 7 contestants, there would be seven possibilities for first place, then six choices for second place, then five choices for third place. So, there are 7 ∗ 6 ∗ 5 = 210 possible ways to accomplish this. The standard notation for this type of permutation is generally n P r or P (n, r). This notation represents the number of ways of allocating r distinct elements into separate positions from a group of n possibilities. In the example above the expression 7 ∗ 6 ∗ 5 would be represented as 7 P 3 or P (7, 3).
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College Algebra & Trigonometry
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c○2018 Richard W. Beveridge This
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4 CONTENTS 2.4 Solution of Rational
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6 CONTENTS 8.2 The Trigonometric Ra
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88 CHAPTER 2. POLYNOMIAL AND RATION
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College Algebra & Trigonometry, 2018a

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