Combinations of a Bracelet?

Question

Dave has 10 poker chips, 6 of which are red and the other 4 of which are white. Dave likes to stack his chips and flip them over as he plays. How many different 10-chip stacks can Dave make if two stacks are not consider distinct if one can be flipped to appear identical to the other?

Answer 1

Let the stack has positions $(1,2,3,4,5,6,7,8,9,10)$

All possibility of stacks is $$\frac{10!}{6! \times 4!}.$$

and the possibility of a stack to be identical while flipping is $$\frac{5!}{3! \times 2!}.$$

No of different stacks is $$\frac{10!}{6! \times 4!} - \frac{5!}{3! \times 2!}.$$