Unit 6

Permutations, Combinations, and The Binomial Theorem

Permutations

Placing items, people, or events in order or in an arrangement is called a permutation.

The order of the objects is important in a permutation. The first position in the arrangement is filled, and then the next position is filled by one of the remaining choices, and so on. When objects or events are distinguishable from one another, a new order makes a new permutation.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»P«/mi»«mi»r«/mi»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«/mstyle»«/math» is the notation for the number of permutations possible when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math» objects are selected from a set of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» objects.

The formula to determine the number of permutations is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»P«/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»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».

Example 4

A class forms a committee that includes a student council representative, a sports council representative, and a fine arts council representative. If there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» students to select from, how many different committees are possible?

Solution

If all students are eligible to be selected for each position, there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» ways to select the student council rep, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»9«/mn»«/mstyle»«/math» ways to select the sports council rep, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math» ways to select the fine arts rep. By the fundamental counting principle,

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»10«/mn»«mo»§#215;«/mo»«mn»9«/mn»«mo»§#215;«/mo»«mn»8«/mn»«mo»=«/mo»«mn»720«/mn»«/mrow»«/mstyle»«/math» ways

In the permutations formula, the total number of students to choose from is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math», and there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» positions to fill.

«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»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»r«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/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»«mmultiscripts»«mi»P«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»10«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»10«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»10«/mn»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»10«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mn»7«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»10«/mn»«mo»§#215;«/mo»«mn»9«/mn»«mo»§#215;«/mo»«mn»8«/mn»«mo»§#215;«/mo»«menclose notation=¨updiagonalstrike¨»«mn»7«/mn»«mo»§#215;«/mo»«mn»6«/mn»«mo»§#215;«/mo»«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»«/menclose»«/mrow»«menclose notation=¨updiagonalstrike¨»«mn»7«/mn»«mo»§#215;«/mo»«mn»6«/mn»«mo»§#215;«/mo»«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»«/menclose»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«mo»§#215;«/mo»«mn»9«/mn»«mo»§#215;«/mo»«mn»8«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»720«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Technology can be used to calculate permutations. Instructions for a TI-83 or TI-84 calculator are shown.

Enter the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» in your calculator.

Then, select the MATH key and the PRB menu.

Then, select «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math»: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»P«/mi»«mi»r«/mi»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«/mstyle»«/math», and enter the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math» on the screen.

Then, press ENTER.