The probability that no two ants will reach the same vertex is 5/256.
Plane, point B and attention. Ants that move to D can come from either E or C. Ants moving to E can come from either B or D of points A and F of the octahedron. Therefore, there are 2 degrees of freedom to decide which ants move to D, 2 degrees of freedom to decide which ants move to E, and 2 degrees of freedom to decide which ants move to A.
So there are 2× 2× 2=8 ways an ant can move to different points.
There are 8 ways for ants to move to different points.
Points B, there are a total of 8 + 4 + 8 = 16 ways Ant can move to her different points. Aligned the square so that point B is defined as point where the ant moved from point A.
An ant can actually move from point A to 4 different points, so there are a total of 4×20=80 ways an ant can move to different points.
Each ant acts independently and has 4 different points to choose from.
So each ant has a 1/4 chance of moving to the desired location. Since he has 6 ants, the probability of appearance is
= 1/4096
So, the desired answer is 80/4096
= 5/256
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