exactly $5$ letters in between $H$ and $N$ 
What is the number of arrangements of the words $\text{HABBANTOTTA}$ in which exactly $5$ letters in between $\text{H}$ and $\text{N}$?

What I tried:
Arranging $\text{H}$ and $\text{N}$ in $2!$ ways. Now selecting $5$ letters from $9$ can as $\displaystyle \binom{9}{5}$ ways and Remaining $4$ letters as $\displaystyle \binom{4}{4}$ and arranging theses $9$ letters as $\displaystyle \frac{9!}{3!\cdot 3!\cdot 2!}$ ways
So total number of ways as $\displaystyle2!\cdot  \binom{9}{5}\cdot \binom{4}{4}\cdot \frac{9!}{3!\cdot 3!\cdot 2!}$
But my answer is wrong.
How do i solve it? Help me please.
 A: Choosing the five letters in the middle isn't as simple as $\binom95$ since there are multiple letters to consider.  Also, the concern with the final four letters isn't which letters they are but how they are spread out before and after the "central" block you just built.
I think the most direct way to think about it is to sort all of the other letters in the $\frac{9!}{3!3!2!}$ ways first.  Then choose whether H or N will be the letter that occurs first, and finally choose between the 5 possible places that that first letter could occur.  
Total arrangements: $2\cdot5\cdot\frac{9!}{3!3!2!}=50400$
A: *

*You are picking the letters to go between the H and the N, and then after that you're freely arranging the 9 letters without regard for which letters are between H and N and which aren't

*You have not taken into account that the H and the N could be placed at different spots in the word (basically, you've only considered the words that start with H or N)


You have $11$ spots to place letters in. There are $5$ possible pairs of spots you can dedicate to the H and N (whichever of them is first can be in anything from the first to the fifth spot, and the second one gets the spot which is six over from there). Then you can arrange the H and the N in $2!$ ways in the two spots you've chosen. Finally, the remaining letters may be arranged in $\frac{9!}{3!\cdot 3!\cdot 2!}$ ways in the remaining 9 vacant spots.
