Combinatorics fun!?

I have had a go at this problem. Sorry, I don't have an answer. There are 18 letters in NASHVILLETENNESSEE and 18 x 17 = 306 possible arrangements of the first N and first E. I drew up an 18 x 18 grid to make it easier to visualise. It seems to me that you have to add up the number of possible arrangements of the 18 letters for each of these 306 cases. Well, I found that I could eliminate all but 126 of them. For example, the first N cannot be in a later position than position # 13, since the other Ns and Ss must come after it. Similarly, the first E cannot be later than position # 13, since the other Es and the T must come after it. So that's 126 different calculations and it would take all day. There may be some further simplification possible that I haven't noticed.

The total number of possible arrangements of the 18 letters if there are no restrictions, is 18!/(2! * 3! * 3! * 5!) = 741,015,475,200. So we know it's less than that.

P.S. I thought up an easier way of doing it. I believe the answer is 308,756,448,000.

I worked out:

(1) the number of possible arrangements of 3 Ns and 3 Ss in which the first N precedes the first S (10);

(2) the number of possible arrangements of the 5 Es and the T in which the first E precedes the T (5);

(3) the number of possible arrangements of the remaining letters (A, H, V, I, L and L) (360).

Lists (1) and (2) both have 6 letters. The number of possible ways of interspersing the 2 lists is 12 choose 6 = 924. The number of possible ways of interspersing this list (the combined lists 1 and 2) with list 3 is 18 choose 6 = 18,564. Thus the number of distinct arrangements of the 18 letters with the first N preceding the first S and the first E preceding the T is

10 x 5 x 360 x 924 x 18,564 = 308,756,448,000