Let's illustrate with smaller #'s first.

Say I have 4 T's and 2 E's, with the same restrictions. I can't have two E's next to each other, so I essentially have this shelf:

_ T _ T _ T _ T _

Where each of the _'s represents a space in which I can place one E.

I have two E's and five spaces, so the number of arrangements is (5 choose 2), or 5!/(2! * 3!). Once I choose the spaces, I'm done - no more arranging.

Since we have 21 T's, we have 22 spaces into which we place our E's. We have 19 E's, so our equation is (22 choose 19), which gives 1540.