Raginsheep
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Assuming you have the letters AA BB CC DD, in how many ways can you arrange them so no two letters of the same type are together?
Perms and Combs Question (1 Viewer)
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Raginsheep
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The answer is 864. Apparently.
grendel
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ok her goes.
the total number of arrangements=(total with no restrictions)-
..........................[(total arrangements with all four letters adjacent)+
.............................(total arrangements with three of the letters adjacent)+
...............................(total arrangements with two of the letters adjacent)+
.................................(total arrangements with one letter adjacent)]
total with no restrictions = 8!/(2!2!2!2!)=2520
total arrangements with all four letters adjacent=4!=24
total arrangements with three of the letters adjacent=[5!/2!-24] X 4C3=144
total arrangements with two of the letters adjacent=[6!/(2!2!)-(24+36+36)] X 4C2=504
total arrangements with one letter adjacent=[7!/(2!2!2!)-(24+36+36+36+84+84+84)] X 4C1=984
TF the total number of arrangements = 2520-(24+144+504+984) = 864
NOTE: all numbers in red are subtracted from the total as these were counted in the previous case(s)
all numbers in the [] are arrangements for one instance only and are multiplied by the numbers in blue to take into account all the different instances.
the total number of arrangements=(total with no restrictions)-
..........................[(total arrangements with all four letters adjacent)+
.............................(total arrangements with three of the letters adjacent)+
...............................(total arrangements with two of the letters adjacent)+
.................................(total arrangements with one letter adjacent)]
total with no restrictions = 8!/(2!2!2!2!)=2520
total arrangements with all four letters adjacent=4!=24
total arrangements with three of the letters adjacent=[5!/2!-24] X 4C3=144
total arrangements with two of the letters adjacent=[6!/(2!2!)-(24+36+36)] X 4C2=504
total arrangements with one letter adjacent=[7!/(2!2!2!)-(24+36+36+36+84+84+84)] X 4C1=984
TF the total number of arrangements = 2520-(24+144+504+984) = 864
NOTE: all numbers in red are subtracted from the total as these were counted in the previous case(s)
all numbers in the [] are arrangements for one instance only and are multiplied by the numbers in blue to take into account all the different instances.
justchillin
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Lesson learnt: i hate probability and counting... 