L6.1 B3 Practice Solutions
Completion requirements
Unit 6
Permutations, Combinations, and The Binomial Theorem
Practice Solutions
Practice 6.1B Solutions
Use these solutions to correct your work. When finished, give yourself a grade using the Practice Assessment rubric.
Pages 534 to 536 questions 1a to 1d, 2, 4a, 4b, 5a to 5c, 6c, 8, 10a, 10b, 12, and 15a to 15c.
a.
The order of the people who shake hands is not important. John and Joe shaking hands is the same as Joe and John shaking hands. This is a combinations problem.
b.
The order of the digits does matter, so this is a permutations problem.
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» cars are being chosen from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»15«/mn»«/mstyle»«/math». The order of selection does not matter. This is a combinations problem.
d.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»6«/mn»«/mstyle»«/math» players are selected from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»18«/mn»«/mstyle»«/math». The order does not matter. This is a combinations problem.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»P«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»5«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mfrac»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»5«/mn»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»60«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»5«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mfrac»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»5«/mn»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»10«/mn»«/mrow»«/mstyle»«/math»
The notation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»P«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»5«/mn»«none/»«/mmultiscripts»«/mstyle»«/math» represents the number of permutations that are possible when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» objects are selected and arranged from a group of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math». Note that order is important when counting permutations. For example, AB is different than BA when permutations are considered.
The notation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»5«/mn»«none/»«/mmultiscripts»«/mstyle»«/math» represents the number of combinations that are possible when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» objects are selected from a group of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math». Note that order is not important when counting combinations.
a.
To choose a group of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» from a total of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math», combinations can be used. A group is being selected and order does not matter.
«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»C«/mi»«mn»4«/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»4«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»4«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»210«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
b.
From «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» employees, the first job can be filled «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» ways.
The second job can be filled «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»9«/mn»«/mstyle»«/math» ways.
The third job can be filled «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math» ways.
The fourth job can be filled «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»7«/mn»«/mstyle»«/math» ways.
Using the fundamental counting principle,
The second job can be filled «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»9«/mn»«/mstyle»«/math» ways.
The third job can be filled «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math» ways.
The fourth job can be filled «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»7«/mn»«/mstyle»«/math» ways.
Using 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»§#215;«/mo»«mn»7«/mn»«mo»=«/mo»«mn»5«/mn»«mspace width=¨0.33em¨/»«mn»040«/mn»«/mrow»«/mstyle»«/math»
or
«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»4«/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»4«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mspace width=¨0.33em¨/»«mn»040«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
a.
AB, AC, AD, BC, BD, CD
When considering combinations, the order does not matter. We did not count BA because AB is the same pair.
When considering combinations, the order does not matter. We did not count BA because AB is the same pair.
b.
Permutations means order matters.
AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC
AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC
c.
There are twice as many permutations as combinations in this problem because each pair of letters can be arranged in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»!«/mo»«/mrow»«/mstyle»«/math» ways.
c.
«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»C«/mi»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd»«mmultiscripts»«mi»C«/mi»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«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»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»6«/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»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mn»2«/mn»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»12«/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»«mn»12«/mn»«/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»«mn»12«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»12«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mi»n«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mi»n«/mi»«mo»§#8722;«/mo»«mn»12«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/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 center right center left¨»«mtr»«mtd»«mfenced»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«/mtd»«mtd»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»n«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«/mtd»«mtd/»«mtd»«mi»n«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The value of «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; therefore, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mo»=«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math» the only solution.
a.
| Left Side | Right Side |
| «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»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»11«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»11«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»11«/mn»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»11«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mn»8«/mn»«mi mathvariant=¨normal¨»!«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/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»C«/mi»«mn»8«/mn»«none/»«mprescripts/»«mn»11«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»11«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»11«/mn»«mo»§#8722;«/mo»«mn»8«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»8«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»11«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«mn»8«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» |
The left side equals the right side.
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» coins are being chosen from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» coins.
«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»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»4«/mn»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
b.
This is a situation that involves cases. Assume that at least one coin must be selected. Let case «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» be that one coin is selected and case «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» be that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» coins are selected. Add the possibilities in each case to determine the total number of possibilities.
«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»C«/mi»«mn»1«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«mo»+«/mo»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mo»+«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Left Side
Right Side «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mo»§#160;«/mo»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«none/»«/mmultiscripts»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mo mathvariant=¨italic¨»§#160;«/mo»«mfrac»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«mo»§#8722;«/mo»«mi»r«/mi»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mstyle»«/math»
Left side = Right side
«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»C«/mi»«mrow»«mi»r«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«mo»+«/mo»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«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»«mfenced»«mrow»«mi»r«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»r«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«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»«/mtd»«/mtr»«mtr»«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»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»r«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»§#8226;«/mo»«mfrac»«mi»r«/mi»«mi»r«/mi»«/mfrac»«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»«mo»§#8226;«/mo»«mfrac»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»r«/mi»«mfenced»«mrow»«mi»r«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced open=¨[¨ close=¨]¨»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfenced»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mo»+«/mo»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»r«/mi»«mo»+«/mo»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»n«/mi»«mi mathvariant=¨normal¨»!«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mi»r«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Right Side «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mo»§#160;«/mo»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«none/»«/mmultiscripts»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mo mathvariant=¨italic¨»§#160;«/mo»«mfrac»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«mo»§#8722;«/mo»«mi»r«/mi»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mi»r«/mi»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mstyle»«/math»
Left side = Right side
a.
Drawing a line from point A to point B is the same as drawing from point B to point A. The order does not matter.
Method 1: Combinations
Method 2: Diagram
Draw all the possible line segments connecting the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» points, and then count the segments.
Method 1: Combinations
There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» points, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» points are chosen for each line segment.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mprescripts/»«mn»5«/mn»«none/»«/mmultiscripts»«mmultiscripts»«mo»=«/mo»«mprescripts/»«mn»2«/mn»«none/»«/mmultiscripts»«mn»10«/mn»«/mrow»«/mstyle»«/math»
Draw all the possible line segments connecting the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» points, and then count the segments.
b.
There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» points, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» points are chosen to make a triangle.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mprescripts/»«mn»5«/mn»«none/»«/mmultiscripts»«mmultiscripts»«mo»=«/mo»«mprescripts/»«mn»3«/mn»«none/»«/mmultiscripts»«mn»10«/mn»«/mrow»«/mstyle»«/math»
c.
The number of triangles is determined by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mprescripts/»«mn»10«/mn»«none/»«/mmultiscripts»«mmultiscripts»«mo»=«/mo»«mprescripts/»«mn»3«/mn»«none/»«/mmultiscripts»«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»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»10«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mn»7«/mn»«mi mathvariant=¨normal¨»!«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»120«/mn»«/mrow»«/mstyle»«/math» because there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» points, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» points are chosen to make a triangle. The order does not matter.
The number of line segments is determined by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«mn»10«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mfrac»«mrow»«mn»10«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»10«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»2«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»10«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mn»8«/mn»«mi mathvariant=¨normal¨»!«/mi»«mn»2«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»45«/mn»«/mrow»«/mstyle»«/math» because there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» points, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» points are chosen to make a line segment. The order does not matter. There are more triangles made than line segments.
The number of line segments is determined by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«mn»10«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mfrac»«mrow»«mn»10«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»10«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»2«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»10«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mn»8«/mn»«mi mathvariant=¨normal¨»!«/mi»«mn»2«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»45«/mn»«/mrow»«/mstyle»«/math» because there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» points, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» points are chosen to make a line segment. The order does not matter. There are more triangles made than line segments.