The notation ${}^{12}P_6$ represents a permutation, which is calculated using the formula:

${}^nP_r = \frac{n!}{(n - r)!}$

Substituting $n = 12$ and $r = 6$:

${}^{12}P_6 = \frac{12!}{(12-6)!} = \frac{12!}{6!}$

Expanding factorials:

$\frac{12!}{6!} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6!}{6!}$

Cancel $6!$ in the numerator and denominator:

$12 \times 11 \times 10 \times 9 \times 8 \times 7 = 665280$

Thus, the final answer is:

$\mathbf{665280}$

Would you like a step-by-step explanation or more details?

Here are five related questions for further exploration:

1. What is the difference between permutations and combinations?
2. How do you calculate ${}^{n}P_{r}$ when given different values of $n$ and $r$?
3. How does factorial notation work, and why is it used in permutations?
4. Can you explain an application of permutations in real life?
5. How would you calculate ${}^{12}C_6$, which represents combinations instead of permutations?

Tip: In permutations, the order matters, while in combinations, it does not.