In how many ways the letters of the word “LOCKDOWN” can be arranged such that N is always between O’s

Question

My attempt :-

I made all the possible cases by fixing the positions of O in word, for example :-

1. O at the 1st place and 3rd place
2. O at the 1st place and 4th place
3. O at the 1st place and 5th place
4. O at the 1st place and 6th place
5. O at the 1st place and 7rd place
6. O at the 1st place and 8th place
7. O at the 2nd place and 4th place
   .
   .
   .
   .
   .

This process seems too ineffecient for solving this problem.

Official answer is given as $\frac{8!}{2!}$ /3 = $6720$

I understand that $\frac{8!}{2!}$ is the total 8 letter rearrangements of the word but why in the solution is the author dividing this by 3 ?

Answer 1

Another way to do it is to first place $-O - N - O -$
and then place the remaining $5$ letters one by one.

The first letter, say $L,$ can be placed in $4$ ways, the next in $-O-L-N-O-\;\;5$ ways, and so on, thus simply

$4\cdot5\cdot6\cdot7\cdot8 = 6720$