# Difference between revisions of "2006 AMC 10A Problems/Problem 25"

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== Problem == | == Problem == | ||

− | A bug starts at one vertex of a cube and moves along the | + | A bug starts at one [[vertex]] of a [[cube (geometry) | cube]] and moves along the [[edge]]s of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal [[probability]] of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? |

<math>\mathrm{(A) \ } \frac{1}{2187}\qquad\mathrm{(B) \ } \frac{1}{729}\qquad\mathrm{(C) \ } \frac{2}{243}\qquad\mathrm{(D) \ } \frac{1}{81}\qquad\mathrm{(E) \ } \frac{5}{243}\qquad</math> | <math>\mathrm{(A) \ } \frac{1}{2187}\qquad\mathrm{(B) \ } \frac{1}{729}\qquad\mathrm{(C) \ } \frac{2}{243}\qquad\mathrm{(D) \ } \frac{1}{81}\qquad\mathrm{(E) \ } \frac{5}{243}\qquad</math> | ||

== Solution == | == Solution == | ||

+ | {{solution}} | ||

== See Also == | == See Also == | ||

*[[2006 AMC 10A Problems]] | *[[2006 AMC 10A Problems]] | ||

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[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||

+ | [[Category:Introductory Combinatorics Problems]] |

## Revision as of 11:23, 30 October 2006

## Problem

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

## Solution

*This problem needs a solution. If you have a solution for it, please help us out by adding it.*