Arranging the letters of INCONVENIENCE so that no C is adjacent to an N As the title indicates, I would like to find the number of ways to arrange the letters of INCONVENIENCE so that no C is adjacent to an N. This is a problem I just made up, and I am interested in finding the easiest way to solve problems like this. (I think the answer is 2,187,360; but I'm not sure.)

As Bhaskar had it, start with $- V - O - I - E - I - E - E -$.
Either both $C$s go in the same slot, leaving four $N$s in the remaining seven slots, or the $C$s go in different slots, leaving four $N$s in the remaining six.
So $$\frac{7!}{3!2!}\left[8{10\choose4}+{8\choose2}{9\choose4}\right]$$