1. How many different necklaces of six beads can be formed using red, blue and green beads? What about $500$-bead necklaces?
2. How many musical scales consisting of $6$ notes are there?
3. How many isomers of the compound xylenol, $\text{C} _6\text{H} _3(\text{CH} _3)_2(\text{OH} )\text{,}$ are there? What about $\text{C} _n \text{H} _{2n+2}\text{?}$ (In chemistry, isomers are chemical compounds with the same number of molecules of each element but with different arrangements of those molecules.)
4. How many nonisomorphic graphs are there on four vertices? How many of them have three edges? What about on $1000$ vertices with $257,000$ edges? How many $r$-regular graphs are there on $40$ vertices? (A graph is $r$-regular if every vertex has degree $r\text{.}$)