Final answer:
In the context of circular permutations, we must consider rotational arrangements as identical. Thus, for 13 different beads on a circular string, the correct number of distinct necklaces is 12!, as we fix one bead and permute the remaining 12.
Step-by-step explanation:
The question is asking how many different necklaces can be created using 13 different beads on a circular string. When we are dealing with circular permutations, we need to take into account the fact that rotations of the same arrangement are considered identical. For a linear arrangement of 13 beads, there would be 13! (13 factorial) different permutations. However, since this is a circular permutation, we consider one bead as a fixed point and arrange the others relative to it, which gives us 12! (12 factorial) permutations because one position is fixed, and we only need to arrange the remaining 12 beads. This accounts for rotations being considered the same arrangement. Therefore, the correct answer is B) 12!.