L6.1 B1 Combinations - Part 3
Completion requirements
Unit 6
Permutations, Combinations, and The Binomial Theorem
Combinations With More Than One Case to Consider
A selection of objects in a group can be made based on one case or another being fulfilled. In the previous example, there was one case; there must be seven Grade 11 students and seven Grade 12 students.
Consider the scenario from Example 2. If the situation had been such that there were to be no more than four Grade 12 students, then each possible case must be considered separately.
In other words, there could have been
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»11«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»12«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» Grade 12 and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»13«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»14«/mn»«/mstyle»«/math» Grade 11s
If one case occurred, the other cases would not. Each case occurs independently of the others.
Hint: Every time the word or is used, the operation of addition is used.
In other words, there could have been
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»11«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»12«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» Grade 12 and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»13«/mn»«/mstyle»«/math» Grade 11s or
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» Grade 12s and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»14«/mn»«/mstyle»«/math» Grade 11s
If one case occurred, the other cases would not. Each case occurs independently of the others.
Hint: Every time the word or is used, the operation of addition is used.
Each case is listed below with addition signs separating them.
The number of ways to select no more than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» Grade 12 students is
Note: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mn»1«/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»14«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mn»15«/mn»«/mrow»«/mstyle»«/math».
The number of ways to select no more than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» 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»«mfenced»«mrow»«mmultiscripts»«mi»C«/mi»«mn»4«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«mo»§#215;«/mo»«mmultiscripts»«mi»C«/mi»«mn»10«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«/mrow»«/mfenced»«mo»+«/mo»«mfenced»«mrow»«mmultiscripts»«mi»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«mo»§#215;«/mo»«mmultiscripts»«mi»C«/mi»«mn»11«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«/mrow»«/mfenced»«mo»+«/mo»«mfenced»«mrow»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«mo»§#215;«/mo»«mmultiscripts»«mi»C«/mi»«mn»12«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«/mrow»«/mfenced»«mo»+«/mo»«mfenced»«mrow»«mmultiscripts»«mi»C«/mi»«mn»1«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«mo»§#215;«/mo»«mmultiscripts»«mi»C«/mi»«mn»13«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«/mrow»«/mfenced»«mo»+«/mo»«mfenced»«mrow»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«mo»§#215;«/mo»«mmultiscripts»«mi»C«/mi»«mn»14«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mn»9«/mn»«mspace width=¨0.33em¨/»«mn»189«/mn»«mspace width=¨0.33em¨/»«mn»180«/mn»«mo»+«/mo»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»113«/mn»«mspace width=¨0.33em¨/»«mn»840«/mn»«mo»+«/mo»«mn»69«/mn»«mspace width=¨0.33em¨/»«mn»615«/mn»«mo»+«/mo»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»890«/mn»«mo»+«/mo»«mn»15«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mn»10«/mn»«mspace width=¨0.33em¨/»«mn»374«/mn»«mspace width=¨0.33em¨/»«mn»540«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Note: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»18«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mn»1«/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»14«/mn»«none/»«mprescripts/»«mn»15«/mn»«none/»«/mmultiscripts»«mo»=«/mo»«mn»15«/mn»«/mrow»«/mstyle»«/math».