I think you could justify at least three different answers to this question, the way it's worded. Obviously you can arrange four things in 4! = 24 orders, but we're arranging charms in a bracelet. A bracelet is circular, so if our charms are A, B, C and D, then is the arrangement ABCD really any different from BCDA if we can just rotate the bracelet so the second arrangement matches the first? It should be a circular permutation question, not an ordinary one. If we must use all four of the charms, then the answer is simply 3! = 6 (because in a circular arrangement, it doesn't matter where you put the first thing - it's only the arrangement of the rest of the things relative to the first one that matters).
But if that's the correct interpretation of the question, then why does the question say the charms are 'removable'? I'd guess we're also supposed to count the arrangements we could make using only 3 of the charms, or 2 of them, or 1 of them. And if that's the question (and if we need to use at least 1 charm, which also isn't clear from the wording of the question), the answer is 24, but purely by coincidence:
Using all 4 charms: we can make 3! circular permutations, or 6 arrangements
Using only 3 charms: we have 4 choices for which charm to omit, then can make 2! circular permutations, so 8 arrangements
Using only 2 charms: we have 4C2 = 6 choices for which charms to use, then can make 1! = 1 circular permutation, so 6 arrangements
Using only 1 charm: we have 4 choices of which charm to use
and the answer is 6+8+6+4 = 24.
It's not at all clear what the question even means, but I'm sure it wasn't intended to be a straightforward permutation question - if that was the intention, it would just ask how many ways we can line up four objects.
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com