L6.1 A2 Permutations - Part 3
Completion requirements
Unit 6
Permutations, Combinations, and The Binomial Theorem
Factorial Notation
When the fundamental counting principle is used with natural numbers up to and including the original value, the process is called taking the factorial.The flag scenario could have been solved by stating there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» factorial ways to arrange the flags to make signals.
The shorthand symbol for factorial is the same as an exclamation point, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»!«/mi»«/mstyle»«/math».
«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»«mo»§#215;«/mo»«mn»3«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»120«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
In general, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mo»=«/mo»«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»«mo»§#8230;«/mo»«mfenced»«mn»1«/mn»«/mfenced»«/mrow»«/mstyle»«/math» where «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».
Note: By definition, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»!«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
How many ways can the letters in word ‘real’ be arranged?
There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» ways to select the first letter in the word, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» ways to select the second letter, then «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» ways, and then «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» way to select the last letter.
Most scientific and graphing calculators include a factorial command. Instructions for a TI-83 or TI-84 calculator are shown.
Enter the natural number on which you wish to perform the factorial function.
Select the MATH key.
Select PRB from the menu.
Select 4: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»!«/mi»«/mstyle»«/math» to take the factorial of that number.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»4«/mn»«mo»§#215;«/mo»«mn»3«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo»=«/mo»«mn»24«/mn»«/mrow»«/mstyle»«/math» ways
Or, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»!«/mi»«mo»=«/mo»«mn»24«/mn»«/mrow»«/mstyle»«/math»
Most scientific and graphing calculators include a factorial command. Instructions for a TI-83 or TI-84 calculator are shown.
Enter the natural number on which you wish to perform the factorial function.
Select the MATH key.
Select PRB from the menu.
Select 4: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»!«/mi»«/mstyle»«/math» to take the factorial of that number.