What is the distribution of the number of boys standing between the leftmost girl and the rightmost girl? 
$10$ Boys and $10$ Girls get ordered in a line.  How is $X$, the number of boys standing between the leftmost girl and the rightmost girl, distributed?

I tried thinking of selecting one place from the $20$ for the leftmost girl, and then selecting k places from the $19$ left for the boys. Or selecting one place from the $19$ left for the rightmost girl. I can't figure how to solve this.
Any help is appreciated.
 A: The total number of placements is $\binom{20}{10}$. The number of placements with $n$ boys between the outermost girls is $(11-n)\binom{8+n}8$, namely $11-n$ ways for the pair of outermost girls to be placed, and $\binom{8+n}8$ ways to place the remaining $8$ girls in the places in between. So the distribution is given by those numbers for $n=0,\ldots,10$ over a common denominator of $\binom{20}{10}=184756$. The respective numerators are $$11,90,405,1320,3465,7722,15015,25740,38610,48620,43758$$
and indeed their sum is $184756$.
A: Between the leftmost girl and the rightmost girl are always 8 girls plus $0$ to $10$ boys.
Now:

*

*If there is no boy between them, the group of $\color{red}{10}$ girls may be in 11 different configurations:
[GGGGGGGGGG]bbbbbbbbbb
b[GGGGGGGGGG]bbbbbbbbb
     ...
bbbbbbbbbb[GGGGGGGGGG]



*If there is 1 boy, the group from the leftmost girl to the rightmost girl will have $\color{red}{11}$ members, and there are only 10 different places for the group as a whole, for example
bb[GGGGGGbGGGG]bbbbbbb

On the other hand, every of 10 possible locations may have that 1 boy in 9 different positions inside a group, so there are 10 × 9 = 90 possibilities.


*If there are 2 boys, the group from the leftmost girl to the rightmost girl will have $\color{red}{12}$ members, and there are only 9 different places for the group as a whole.
But every of 9 possible locations may have those 2 boys in $\binom{10} 2$ = 45 different positions inside a group, so there are 9 × 45 = 405 possibilities.
So the distributions of boys between the leftmost and the rightmost girl is for now:




Boys
Occurrences
                                                                                                  




0
11



1
90



2
405





And you may continue ...
