A bug starts at one vertex of a cube and moves along the edges of the

Question

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?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

shuvodip04 · Answer

The bug is a one vertex of the cube. So remaining vertex to be visited 7. So the bug has to visit a new vertex in each of the the subsequent moves. When he is at a vertex he can select one of the edges in 3 ways.

So probability of selecting an edge = 1/3

So the probability that the bug will visit the every vertex once in the next seven moves =  (1/3)^7  =1/2187

Archit3110 · Answer

total paths for the bug to move 3^7 = 2187
no of ways bug can move = 3*3*2 = 18
so P ; 18/2187 ; 2/243
IMO C

iPrasad · Answer

Pls explain highlighted

Archit3110 · Answer

iPrasad 
draw a cube ; so as to cover max distance in a cube i.e opposite sides diagonal side the bug will start from one of the vertices it can move in 3 directions ; then bug can move in 2 directions ; after which again bug will have 3 options to move 
so total max ways a bug can move w/o coming back to vertice already crossed is 3*2*3 ; 18

sekharm2389 · Answer

Hi Bunuel,

Please provide the official answer and the explanation.

Thanks
Sekhar M

Shobhit7 · Answer

Total paths available for the bug= 3 per vertex^7moves = 3^7
Favorable paths in order to visit each vertex in these seven moves= 3*2*2*1*1*1*1 = 12
First vertex= 3 options
Second vertex= 2 options
Third vertex= 2 options
Fourth to seventh vertex= 1 option each

Probability = 12/3^7 = 4/729 is the answer IMO, but none of the options have this ans choice.

Mohammad Ali Khan · Answer

the number of ways bug can move ..can you explain the movement keeping in mind the condition stated?? the way you got 18 is not clear to me.

Archit3110 · Answer

@MohammadAliKhan
draw a cube ; so as to cover max distance in a cube i.e opposite sides diagonal side the bug will start from one of the vertices it can move in 3 directions ; then bug can move in 2 directions ; after which again bug will have 3 options to move 
so total max ways a bug can move w/o coming back to vertice already crossed is 3*2*3 ; 18

neha283 · Answer

Hi  Archit3110  ,

To maximize the number of vertex visited, when the bug is on 3rd vertex, it does have 3 possible options but if it chooses to go back to vertex from which it came then the number won't be maximized. so, shouldn't it be 3*2*2?  Bunuel 
Where am i going wrong?

ryanstevengoff · Answer

Hello,

I believe the 18 originated from the initial possibilities of movement. The bug can go in “3” directions and the probability of any future move is defined by “3!” Thus resulting in 3*3! Which is 3*3*2*1=18.

Posted from my mobile device

neha283 · Answer

Hi  pushpitkc , 
could you help me with the solution of this problem?

TIA

auradediligodo · Answer

Hi,
Hi tried to solve the problem assigning to each step a probability.
as for my solution the probability is: (3/3)*(2/3)*(2/3)*(2/3)*(1/3)*(1/3)*(1/3)=8/729
Can someone explain to why my reasoning is wrong?

samyak27 · Answer

the bug can move from its arbitrary starting vertex to a neighboring vertex in 3 ways. After this, the bug can move to a new neighbor in 2 ways (it cannot return to the first vertex). The total number of paths is 3^7/2187. Therefore, the probability of the bug following a good path is equal to 6x/2187 for some positive integer x. The only answer choice which can be expressed in this form is 2/243.

RBGO · Answer

there are 3 vertex of the box which you can choose 3 lines and there are 4 vertex for you can move 2 lines, and one 1 vertex you already taken...
soo.. no. ways you can move to the vertex with respect to your reference. is equal to = 3(3C1) + 4(2C1) + 1 = 18
and total ways you can move is 3^7 = 2187
so the probability is = 18/2187 = 2/243

bumpbot · Answer

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A bug starts at one vertex of a cube and moves along the edges of the

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