Lance has a regular heptagon (7-sided figure). How many distinct ways can he label the vertices of the heptagon with the letters in OCTAGON if the N cannot be next to an O? Rotations of the same labeling are considered equivalent.
^(add on to my first post) I found that there are 7!/2! ways to label the heptagon considering that there are two O's in "OCTAGON". Since there are 4! = 24 ways where the O's are next to an N, so is the answer 7!/2! - 24?
Am I on the right track?
I answered this a little while ago.
It is normal to assume that rotations are the same.
Put the N anywhere.
now there are 4 places where the os can go. 4C2 = 6
Now there is 4 spots not taken and 4 letters to put into them, that is 4! ways.
So the answer is 6*4! = 144
Why don't you multiply 144*7 at the end of the problem to consider the $N$