Mock AIME 6 2006-2007 Problems/Problem 11

Problem

Each face of an octahedron is randomly colored blue or red. A caterpillar is on a vertex of the octahedron and wants to get to the opposite vertex by traversing the edges. The probability that it can do so without traveling along an edge that is shared by two faces of the same color is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

case 0 blues: Total number of octahedrons with the condition = 0

case 1 blue: Total number of octahedrons with the condition = 0

case 2 blues: Total number of octahedrons with the condition = $\binom{8}{2}-2\binom{4}{2}-2=14$

case 3 blues: Total number of octahedrons with the condition = $\binom{8}{3}-2\binom{4}{3}=48$

case 4 blues: Total number of octahedrons with the condition = $\binom{8}{4}-2\binom{8}{4}=68$

case 5 blues: Total number of octahedrons with the condition = Total from case 3 = 48

case 6 blues: Total number of octahedrons with the condition = Total from case 2 = 14

case 7 blues: Total number of octahedrons with the condition = Total from case 1 = 0

case 8 blues: Total number of octahedrons with the condition = Total from case 0 = 0

Gran Total $=14+48+68+48+14=192$

Probability = $\frac{192}{2^8}=\frac{192}{256}=\frac{3}{4}$

$m=3$ , $n=4$ , $m+n=3+4=\boxed{7}$

~Tomas Diaz. orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

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Mock AIME 6 2006-2007 Problems/Problem 11

Problem

Solution