In how many ways can the word isosceles be rearranged if the two e must not come together

This is very interesting, the es are reasy enough to factor in, but you need to consider the s's, which could be in different positions but form the same letter sequence. I'm not entirely sure how to form this. Hopefully someone else can step in.

Do you have any ideas how to go forward? Have you done any work on it so far?

If you show that you have done some work on it, then we will be much happier to help you :)

The good thing to do here is to count the number of ways the e's CAN be grouped together and subtracting it from the total number of arrangements.

The word ISOSCELES can be arranged in \frac{9!}{3!2!} ways.

To count the number of ways the e's can be side-by-side is to tie them together and arrange has if they're one letter. That can be done in \frac{8!}{3!} ways.

If e's are not side by side:

\frac{9!}{3!2!}-\frac{8!}{3!}