SOLUTION: Not including the identity transformation, the eleven transformations that preserve the regular hexagon shown are counterclockwise rotations by $60^\circ,$ $120^\circ,$ $180^\circ,

Question 1209329: Not including the identity transformation, the eleven transformations that preserve the regular hexagon shown are counterclockwise rotations by $60^\circ,$ $120^\circ,$ $180^\circ,$ $240^\circ$ and $300^\circ$ and reflections across the six dashed lines shown. Kristina randomly picks six transformations $T_1,$ $T_2,$ $T_3,$ $T_4$, $T_5$ and $T_6,$ with replacement, from this set of eleven. She performs these six transformations on the hexagon, in succession. The probability that the point $P$ is transformed to each of the hexagon's six vertices exactly once during this process is $\dfrac{k}{11^6}.$ What is the value of $k\,?$

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
To solve this problem, we need to consider the transformations that can move point P to each vertex exactly once.
The value of k in the probability $\frac{k}{11^6}$ can be calculated as follows:
1. There are 6! = 720 ways to arrange the six vertices in a sequence.
2. For each sequence, we need to count the number of ways to choose transformations that achieve that sequence:
- The first transformation can be any of the 11 that moves P to the first vertex in the sequence.
- The second transformation must be one of the 2 that moves the current position to the second vertex.
- Each subsequent transformation also has 2 choices to move to the next vertex.
3. Therefore, for each sequence, there are 11 * 2^5 = 352 ways to choose the transformations.
4. The total number of favorable outcomes is thus 720 * 352 = 253,440.
Therefore, k = 253,440.
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
.
Not including the identity transformation, the eleven transformations that preserve
the regular hexagon shown are counterclockwise rotations by 60°, 120°, 180°, 240° and 300°
and reflections across the six dashed lines shown. Kristina randomly picks six transformations
T_1, T_2, T_3, T_4, T_5 and T_6, with replacement, from this set of eleven. She performs
these six transformations on the hexagon, in succession. The probability that the point P
is transformed to each of the hexagon's six vertices exactly once during this process is k/11^6}.
What is the value of k ?
~~~~~~~~~~~~~~~~~~~~~~~~~

In my view, this problem, as formulated in the post, is FATALLY DEFECTIVE and is not a mathematical problem, at all.

I will point only two defects in its formulation, but they just make it totally invalid.

First my objection is that "the point $P$" is not defined in the post, despite of using an article "the".

Second objection is that the form k/11^6 can represent ANY number and does not represent any special/specific number,
so, mathematically, this condition that " the probability . . . is k/11^6 " does not determine a concrete number
and, HENCE, defines NOTHING and is EMPTY.

I have many other objectives against this formulation, but these two are just enough to kill this " problem " to the death.

The solution provided to this problem by AI is soup of words without any relevant interior logic in it.

The impression after reading this " solution " is as if the system tries to sell nonsense
under the guise of Artificial Intelligence.

I made similar notices many times, so you just should have understanding that the current version
of your AI is not able to recognize false Math problems and to distinct them from proper Math problems.

So, when a false/(fatally defective) Math problem comes, the system tries to treat
it as a proper Math problem, instead of rejecting it.

From my view, it is the weakest point of the current version of your AI.

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