L6.1 A2 Permutations - Part 5
Completion requirements
Unit 6
Permutations, Combinations, and The Binomial Theorem
A video game designer wishes for a gamer to select two options from a list, which will lead the gamer to one of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»56«/mn»«/mstyle»«/math» possible outcomes. How many options should the list include?
In this case, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» represents the number of options in the list, which is unknown, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math» represents the number of selections to be made, which is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math». There are a total of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»56«/mn»«/mstyle»«/math» possible permutations.
The variable «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» shows up twice in the equation and is part of a factorial both times. How can we eliminate the factorials?
A factorial can be expanded several different ways. For example,
Similarly,
To solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»56«/mn»«mo»=«/mo»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math», expand the larger factorial until a factorial in the numerator matches a factorial in the denominator.
The resulting equation is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mn»56«/mn»«/mrow»«/mstyle»«/math», which can be solved algebraically.
The only permissible value is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mo»=«/mo»«mn»8«/mn»«/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» is a natural number. Recall «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» represents the number of choices the gamer is presented with from which he must choose «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math».
The gamer has «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math» choices from which he will select «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math», and the result will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»56«/mn»«/mstyle»«/math» permutations.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mmultiscripts»«mi»P«/mi»«mn»2«/mn»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«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»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»56«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The variable «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» shows up twice in the equation and is part of a factorial both times. How can we eliminate the factorials?
A factorial can be expanded several different ways. For example,
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mn»5«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mo»§#215;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mo»§#215;«/mo»«mn»4«/mn»«mo»§#215;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mo»§#215;«/mo»«mn»4«/mn»«mo»§#215;«/mo»«mn»3«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Similarly,
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
To solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»56«/mn»«mo»=«/mo»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math», expand the larger factorial until a factorial in the numerator matches a factorial in the denominator.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mn»56«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«menclose notation=¨updiagonalstrike¨»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/menclose»«/mrow»«menclose notation=¨updiagonalstrike¨»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/menclose»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The resulting equation is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mn»56«/mn»«/mrow»«/mstyle»«/math», which can be solved algebraically.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mi»n«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»56«/mn»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mi»n«/mi»«mo»§#8722;«/mo»«mn»56«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»8«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»7«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left center left¨»«mtr»«mtd»«mi»n«/mi»«mo»-«/mo»«mn»8«/mn»«mo»=«/mo»«mn»0«/mn»«/mtd»«mtd»«mi»or«/mi»«/mtd»«mtd»«mi»n«/mi»«mo»+«/mo»«mn»7«/mn»«mo»=«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»n«/mi»«mo»=«/mo»«mn»8«/mn»«/mtd»«mtd/»«mtd»«mi»n«/mi»«mo»=«/mo»«mo»-«/mo»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left center left¨»«mtr»«mtd»«mi»n«/mi»«mo»-«/mo»«mn»8«/mn»«mo»=«/mo»«mn»0«/mn»«/mtd»«mtd»«mi»or«/mi»«/mtd»«mtd»«mi»n«/mi»«mo»+«/mo»«mn»7«/mn»«mo»=«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»n«/mi»«mo»=«/mo»«mn»8«/mn»«/mtd»«mtd/»«mtd»«mi»n«/mi»«mo»=«/mo»«mo»-«/mo»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The only permissible value is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mo»=«/mo»«mn»8«/mn»«/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» is a natural number. Recall «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» represents the number of choices the gamer is presented with from which he must choose «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math».
The gamer has «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math» choices from which he will select «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math», and the result will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»56«/mn»«/mstyle»«/math» permutations.