Arranging Question?

Question

In how many ways can you arrange the letters of the word think so that the t and the h are seperated by at least one other letter.

Thanks for the help!

Answer 1

The number of ways to place the t and h are...
T_H__
T__H_
T___H
_T_H_
_T__H
__T_H
and these same six arrangements with T and H switched.

So, there 12 ways to place the T and H. This leaves you with three places to place the other letters. There are 6 ways to order 3 items, so the total number of arrangements in which T and H are not adjacent is:
12*6 = 72

Answer 2

It is easier to find the number of ways of arranging such that they will be together, and subtract that from the total number of ways of arranging the letters.
t and h are together in (4!)(2!) ways.
and the total number of ways of arranging the letters is 5!.
subtract to get your answer.