Unit 6

Permutations, Combinations, and The Binomial Theorem

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Part 6.1B corresponds to section 11.2, starting on page 528 of your Pre-Calculus 12 textbook.

To combine means to put things together. In a mathematical sense, combinations are ways to select a group from a larger group. The order of the objects selected within the group is not important, but the category of the objects selected is. Determining the number of ways «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» girls and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» boys can be selected from a classroom that consists of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» boys and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»12«/mn»«/mstyle»«/math» girls is an example of a problem that can be solved using combinations. We are looking for a combination of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» girls and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» boys. The order of the selection does not matter.

The formula for calculating the number of combinations is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«mo»=«/mo»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» is the number of objects in the large group from which objects are selected.

Note that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»N«/mi»«/mrow»«/mstyle»«/math» because «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» represents the total number of objects from which a category or group is being selected. «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»§#8804;«/mo»«mi»r«/mi»«mo»§#8804;«/mo»«mi»n«/mi»«/mrow»«/mstyle»«/math», because it represents the number of objects selected in a specified category or group from a total of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» objects.