A necklace is a circular arrangement of beads which can be rotated and flipped without being considered a different arrangement. For example, there is only one necklace strung together from three...
A necklace is a circular arrangement of beads which can be rotated and flipped without being considered a different arrangement. For example, there is only one necklace strung together from three beads where all beads have different colors. a) How many different necklaces with N beads can be made with N distinct colors? b) How many different necklaces can you string together if you have to use 5 beads and up to 2 colors? c) Now suppose we can only rotate necklaces, but not flip them, so rotations are considered the same arrangement but flips are considered different. Can you count the different necklaces with N beads that can be made from a supply of 2 types of beads, given that N is prime and the 2 types need not both be used?
A necklace is a circular arrangement of beads which can be rotated and flipped without being considered a different arrangement. For example, there is only one necklace strung together from three beads where all beads have different colors. a) How many different necklaces with N beads can be made with N distinct colors? b) How many different necklaces can you string together if you have to use 5 beads and up to 2 colors? c) Now suppose we can only rotate necklaces, but not flip them, so rotations are considered the same arrangement but flips are considered different. Can you count the different necklaces with N beads that can be made from a supply of 2 types of beads, given that N is prime and the 2 types need not both be used?