- #1
Amad27
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The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthling must occupy chair $15$, Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $N(5!)^3$. Find $N$.
We have a race: $M_1 M_2 M_3 M_4 M_5$, the number of arrangements within a race $5!$ arrangements.
Not Possible Arrangements: $EM, VM, VE$.
We start with $M$ and end with an $E$.
If we look carefully, we can put: $MEV$ or $MMEEVV$ but not $MVE$
So it is possible to have:
$MMMMMEEEEEVVVVV$ or $MEVMEVMEVMEVMEVMEV$
But this isn't helpful at all.
We have a race: $M_1 M_2 M_3 M_4 M_5$, the number of arrangements within a race $5!$ arrangements.
Not Possible Arrangements: $EM, VM, VE$.
We start with $M$ and end with an $E$.
If we look carefully, we can put: $MEV$ or $MMEEVV$ but not $MVE$
So it is possible to have:
$MMMMMEEEEEVVVVV$ or $MEVMEVMEVMEVMEVMEV$
But this isn't helpful at all.