A linear chain is made of 20 identical links. Each link can be made in 7 different colors. How many # physically # different chains are there?
For each of 20 links, there are 7 choices, each time the choice is independent of previous choices, so we can take product.
Total number of choices =
But since chain can be reversed, we need to count distinct sequences.
First, we count number of symmetric sequences: i.e last 10 links take the mirror image of first 10 links.
Number of symmetric sequences = number of ways so select first 10 links =
Except for these symmetric sequences, the non symmetric sequences can be reversed to produce a new chain. This means that only half of non-symmetric sequences are unique.
Number of unique sequences = (Number of non-symmetric)/2 + Number of symmetric sequences