732 CHAPTER 11 Probability In Section 11.7, we considered problems in which selections of objects were made where the order of the selections was important. These problems involved permutations. In this section, we will consider problems in which selections of objects are made where the order of the selections is not important. Two such problems were presented in the section opener. These problems involve combinations. For example, a b c , , and b c a , , are two different permutations of the same letters because the ordering of the three letters is different. The letters a b c , , and b c a , , represent the same combination of letters because the same letters are used in each set. However, the letters a b c , , and a b d , , represent two different combinations of letters because the letters contained in each set are different. Isaac is the faculty advisor for the Brain Bowl team at Northeast Texas Community College. For an upcoming tournament, Isaac selects a team of four players from six eligible students. How many different teams of four players are possible? After the tournament, Isaac takes the team out for pizza. They order an extra-large pizza and must choose three toppings from a list of 11 toppings. How many different three-topping pizzas are possible? In this section, we will answer these and similar questions that involve forming an arrangement of objects (or people) without regard to the order of the objects. Such arrangements are called combinations. SECTION 11.8 Combinations LEARNING GOAL Upon completion of this section, you will be able to: 7 Solve problems involving combinations. Why This Is Important Combinations are used in many branches of mathematics including probability, statistics, analysis, operations research, and abstract algebra. Mathematical applications that involve combinations play a role in data storage and data mining. Combinations are also used in chemistry, physics, and biology. A basic understanding of combinations may be helpful to you in future mathematics and science classes. Definition: Combination A combination is a distinct group (or set) of objects without regard to their arrangement. Example 1 Permutation or Combination The Earth Club at Joliet Junior College has members Jaspar, Carla, Opal, David, and Malcolm. Determine whether the description given below regarding a selection involving the Earth Club members describes a permutation or a combination. a) The club will select a president and a treasurer. b) Of the five club members, two will be attending a Sierra Club meeting together. Solution a) Since the president’s position is different from the treasurer’s position, the order in which the selection is made is important. For example, Jaspar as president and Carla as treasurer is different from Carla as president and Jaspar as treasurer. Since the order of the selection is important, this is a permutation. VGstockstudio/Shutterstock
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