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Permutations and Combinations. Example #1. In how many ways can 4 of the 9 members of the U.S. Supreme Court enter a room, one at a time?. The problem in Example #1 was solved by applying the multiplication principle.
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Example #1 • In how many ways can 4 of the 9 members of the U.S. Supreme Court enter a room, one at a time?
The problem in Example #1 was solved by applying the multiplication principle. • We are about to define permutations and we will see that there is another way of solving this problem.
Definition • A permutation of r objects selected from a set of n different objects is an arrangement of r of the n objects in a specific order. • The number of permutations of r objects selected from a set of n different objects, denoted by nPr, is given by • Here n!, read n factorial, is the product of the first n positive integers. • For instance, 3!=3⋅2⋅1= 6 and 4! != 4⋅ 3⋅2⋅1= 24 • We define 0!= 1.
Example #2 • Use the permutation formula to find in how many ways can 4 of the 9 members of the U.S. Supreme Court enter a room, one at a time?
Good News! • We can use a graphing calculator to compute
Example #3 • A TV network has 20 different shows available for Friday nights. You are asked to schedule 6 of them. • How many different sequences of 6 shows are possible?
Can you see why the following is NOT a permutation problem? • Think about it seriously before moving on! What’s different! • You have 15 compact discs. Your girlfriend wants to borrow 3 of them. • In how many different ways can she select 3 CDs?
Definition • A combination of r objects selected from a set of n different objects is a selection of r of the n objects with order disregarded. • The number of combinations of r objects selected from a set of n different objects, denoted by nCr, is given by • Here n!, read n factorial, is the product of the first n positive integers.
Example #4 • You have 15 compact discs. Your girlfriend wants to borrow 3 of them. • In how many different ways can she select 3 CDs?
Another Good News! • We can use a graphing calculator to compute
Remarks • You will find it useful to practice distinguishing combination from permutation problems. • Remember that • in permutation problems: order is indicated in one way or another. • in combination problems, we have only unordered selections. • Phrases like arrange, schedule, line up, and so on, suggest the existence of some order.
Example #5 • A 5-person committee is to be formed from a group of 10 female and 7 male executives. • How many 5-person committees are possible? • How many of those committees contain 3 females and 2 males? • How many of those committees contain no male? • How many of those committees contain at most 1 male?
Example #6 • The U.S. Senate consists of 100 members. • The Senators need to elect a 5-person committee, the Federal Response to Katrina Committee. • The Senators also need to elect 5 officers (chairperson, first vice chairperson, second vice chairperson, secretary, assistant secretary) to form a Space Program Advisory Board. • In how many different ways can the FRKC be formed? • How many different slates of candidates are possible for the SPAB?
Note that order is irrelevant in forming the FRKC. • However, order does count with the slates of candidates to be elected to the SPAB. • The number of possible different FRKCs • We select 5 Senators from the 100 U.S. Senators • So r=5 and n=100. • So there are ₁₀₀C₅ =75,287,520 ways that the FRKC may be formed.
The number of possible different SPABs • We select 5 Senators from the 100 U.S. Senators • So r=5 and n=100. • So there are ₁₀₀P₅ =9,034,502,400 ways that the SPABs may be formed. You might have expected a large number, perhaps not that large! End of Example #6
