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 $k\,?$

To solve the problem, let's first analyze the transformations that can be performed on a regular hexagon. The eleven transformations include:

- Six rotations: $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ$, and the identity transformation $0^\circ$.
- Six reflections across lines that pass through vertices or midpoints of opposite sides.

However, the identity transformation does not count in the problem, leaving 10 transformations (5 rotations and 5 reflections).

Kristina randomly picks six transformations $T_1, T_2, T_3, T_4, T_5, T_6$ from these 10 transformations (with replacement), and performs them in succession. We are to find the probability that the point $P$ is transformed to each of the hexagon's six vertices exactly once during this process.

### Understanding the Problem
The problem is asking for the number of sequences of transformations that move point $P$ to all six vertices exactly once, in any order.

Since we are transforming $P$ to each vertex exactly once, the sequence of transformations must be a permutation of the hexagon’s six vertices. The six vertices can be permuted in $6!$ different ways.

### Analyzing the Transformations
For each permutation of vertices, there must be a corresponding sequence of transformations that can move $P$ accordingly. However, not all sequences of transformations will result in valid permutations due to the properties of the hexagon's symmetries.

A key insight here is that the product of all six transformations must be the identity transformation (since starting from $P$, we end up with $P$ back at its original position after visiting all six vertices). This constraint heavily restricts the sequences of transformations that are possible.

### Counting Valid Sequences
The identity transformation constraint implies that the six transformations must correspond to an element of the alternating group $A_6$ on 6 elements. There are $60$ elements in $A_6$, but we need to determine how many correspond to actual sequences of transformations.

Since there are 11 choices for each transformation, and the transformations must form a sequence where each vertex is visited exactly once, the number $k$ (the numerator of the probability) is simply the number of valid sequences that can occur, which is exactly $60$.

Thus, the total number of valid sequences is 60, corresponding to the number of distinct permutations of the vertices modulo rotation. So, we get:

\[
k = 60
\]

Therefore, the probability is $ \frac{k}{11^6} $ where $ k = 60 $.

### Final Answer
\[
\boxed{60}
\]

Explore the solution to a probability problem involving transformations on a regular hexagon. Understand how permutations and group theory concepts are applied to calculate the probability. Suitable for advanced high school students.

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