L6.1 B1 Combinations - Part 2
Completion requirements
Unit 6
Permutations, Combinations, and The Binomial Theorem
Technology can also be used to solve problems involving combinations.
Determine the number of ways a soccer team can be selected from a group of
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»15«/mn»«/mstyle»«/math» Grade 11 students and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»18«/mn»«/mstyle»«/math» Grade 12 students if
a.
there are to be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»14«/mn»«/mstyle»«/math» players on the team.
b.
there are to be an equal number of Grade 11 students and Grade 12 students on the team.
a.
There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»15«/mn»«/mstyle»«/math» Grade 11 students and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»18«/mn»«/mstyle»«/math» Grade 12 students. 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¨»«mrow»«mn»15«/mn»«mo»+«/mo»«mn»18«/mn»«mo»=«/mo»«mn»33«/mn»«/mrow»«/mstyle»«/math».
From «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»33«/mn»«/mstyle»«/math» students, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»14«/mn»«/mstyle»«/math» can be chosen for the team.
Enter «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mn»14«/mn»«none/»«mprescripts/»«mn»33«/mn»«none/»«/mmultiscripts»«/mstyle»«/math» in your calculator using the following steps.
There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»818«/mn»«mspace width=¨0.33em¨/»«mn»809«/mn»«mspace width=¨0.33em¨/»«mn»200«/mn»«/mrow»«/mstyle»«/math» ways to select a team of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»14«/mn»«/mstyle»«/math» students from a total of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»33«/mn»«/mstyle»«/math» students. Wow! If it was based on random selection, and not on skill, that is indeed a lot of ways to make a team.
From «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»33«/mn»«/mstyle»«/math» students, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»14«/mn»«/mstyle»«/math» can be chosen for the team.
«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»33«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»r«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»14«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Enter «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mn»14«/mn»«none/»«mprescripts/»«mn»33«/mn»«none/»«/mmultiscripts»«/mstyle»«/math» in your calculator using the following steps.
To use a TI-83 or TI-84 calculator, enter the total number of students.
Then, press the MATH key, and select the PRB menu.
Choose 3: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«/mstyle»«/math»
Choose 3: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«mi»n«/mi»«none/»«/mmultiscripts»«/mstyle»«/math»
Then, enter the number of students on the team,
which is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»14«/mn»«/mstyle»«/math», and press ENTER.
b.
Now, there is a restriction on the ways to select the team members.
There are to be an equal number of Grade 11 and Grade 12 students. There will be seven Grade 11 and seven Grade 12 students on the team.
First, determine the number of ways to choose seven Grade 11 students from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»15«/mn»«/mstyle»«/math». Then, determine the number of ways to choose seven Grade 12 students from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»18«/mn»«/mstyle»«/math».
The number of ways to select seven Grade 11 students and seven Grade 12 students is
There are to be an equal number of Grade 11 and Grade 12 students. There will be seven Grade 11 and seven Grade 12 students on the team.
First, determine the number of ways to choose seven Grade 11 students from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»15«/mn»«/mstyle»«/math». Then, determine the number of ways to choose seven Grade 12 students from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»18«/mn»«/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»7«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»15«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»15«/mn»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»7«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mspace width=¨0.33em¨/»«mn»435«/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»C«/mi»«mn»7«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»18«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»18«/mn»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»7«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»31«/mn»«mspace width=¨0.33em¨/»«mn»824«/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»C«/mi»«mn»7«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»18«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«mrow»«mfenced»«mrow»«mn»18«/mn»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»!«/mi»«mn»7«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»31«/mn»«mspace width=¨0.33em¨/»«mn»824«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The number of ways to select seven Grade 11 students and seven Grade 12 students is
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mo»=«/mo»«mn»6«/mn»«mspace width=¨0.33em¨/»«mn»435«/mn»«mo»§#215;«/mo»«mn»31«/mn»«mspace width=¨0.33em¨/»«mn»824«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mn»204«/mn»«mspace width=¨0.33em¨/»«mn»787«/mn»«mspace width=¨0.33em¨/»«mn»440«/mn»«mi mathvariant=¨normal¨».«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»