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.

Guest Jun 25, 2020

#1**0 **

^(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?

Guest Jun 25, 2020

#2**+1 **

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

Melody Jun 25, 2020