Maths question from the chapter of combinations

Question

How many different words can be formed with word TOMORROW taking all at a time such that all vowels don't occur together??

Answer 1

Some hints.

1. Calculate the number of all possible words you can make with the letters. (Multinomials will help.)
2. Subtract out the words that have $OOO$ in them. (How many places can $OOO$ occur? Then, how many ways can the rest of the letters fill in the remaining $5$ slots?)

Spolier

From multinomials, the number of words is $8!/(3!2!)$. There are $6$ places to put $OOO$, and $5!/2!$ ways to fill in $TMRRW$. So, the answer is $\frac{8!}{3!2!} - 6 \cdot \frac{5!}{2!}$.