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6-4a - Chapter 6.4 Permutations and Combinations Example...

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Chapter 6.4: Permutations and Combinations Example: Suppose six friends wish to line up for a photo. How many ways can this be done? Using the rule of product, there are six tasks to complete: who is the first person from the left, who is the second, the third, fourth, fifth, and sixth. There are six choices for the first person. Then, there will be five choices for the second person, four for the third person, and so on. 6 × 5 × 4 × 3 × 2 × 1 = 720 —– —– —– —– —– —– 1st 2nd 3rd 4th 5th 6th There are 720 different arrangements. Example: Suppose only five friends wish to line up for a photo. How many different arrangements are there? We solve the problem in the same way as before: 5 × 4 × 3 × 2 × 1 = 120 —– —– —– —– —– 1st 2nd 3rd 4th 5th There are 120 different arrangements. Notice that when we wished to arrange 6 people that it could be done 6 × 5 × 4 × 3 × 2 × 1 ways, and that when we wished to arrange 5 people, it could be done 5 × 4 × 3 × 2 × 1 ways. In general, arranging n objects, can be done n × ( n - 1) × · · · × 2 × 1 ways. These products have a special name, and a symbol. 1

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Suppose there are n distinct objects we wish to arrange. The number of different arrangements is n !, which we read as “ n factorial ”. The symbol n ! means the following: n ! = n × ( n - 1) × · · · × 2 × 1. n ! is defined for positive integer values of n . We also define 0! = 1. This tool helps us tackle certain counting problems quickly. You should note that your calculator contains a factorial button, to help speed up the solution of problems. Example: How many different ways are there to arrange the letters in the word “OBJECTS”? Since there are seven distinct letters in the word, then there are 7! = 5040 different arrangements. Example: Ten people get in line for a movie. How many different ways are there for them to get in line?
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