How many distinctively different necklaces are possible if you use 0 to 3 beads out of 5 different beads given?

Question

Today in my IB Math II HL/AP Calculus BC class we were going over IB Math I topics and did the given problem. So you would calculate the number of necklaces for a 0, 1, 2, and 3 bead necklace. I used $5C0+5C1+5C2(2!)(1/2)+5C3(3!/3)(1/2)=26 to find my answer. In a two bead necklace it would be multiplied by 2 because of the two different combinations, but then is cancelled out because a necklace is circular. However, my teacher said that a two bead necklace cannot exist, and therefore did not divide by 2, saying that a necklace A-B and B-A are two different necklaces. He said that two points cannot make a circular shape and must be treated as linear. Maybe he is right?? Therefore he gave the answer 36 because he did not divide the 5C2 term by 2. Thank you in advance, I hope I can find an answer to this problem.