In how many ways can the letters of the word COMMON be arranged so that neither two M's nor two O's are together?

why can't we do like this:Consider two M's together(I) and two O's together(II),total-(I+II) .

Let the no. of ways in which both M's and O's are together be iii.Then,We can see that i and ii both include iii in them.I mean,In the cases when M's are together,there will be some cases in which both M's and O's are together.Similarly,when O's are together,there will be some cases in whuch both O's and M's are together.So,we are subtracting these cases two times and hence,need to add them once.So,the answer is

(Total)-(I+II)+III.

In this case,Total=6!/2!2! ,I=5!/2! ,II=5!/2!  ,III=4!

Hence,the answer is (6!/4)-5!+4!=180-120+24=84 ways.

Hope you got it.Is my answer correct?