A volunteer tutor wants to know HOW to solve this problem. It is not
necessary to give the answer if you have explained it so that a
student can get it...
A necklace has eight gems. Two are diamonds. How many ways can the
gems be arranged so that the diamonds are never adjacent?

Assuming the other eight gems are distinct, compute the total number
of ways and subtract the number of bad configurations.

Are the gems all distinct (i.e. are we looking at arrangements of the
letters ABCDEFGH or arrangements of AAAAAADD)? Are the diamonds
distinct (i.e. are we looking at arrangements of AAAAAADd or
AAAAAADD)? Are two arrangements considered to be the same if you can
get one by flipping over the necklace (i.e. so ABCDEFGH is the same as
HGFEDCBA)? Are two arrangements considered to be the same if you can
get one by pushing some of the beads all the way around the necklace
(so ABCDEFGH is the same as DEFGHABC)? All of these possibilities give
rise to non-trivial questions.