1 a)
in solution,
typo:
it should be:
numinternal(...)= 1+ \sum_ ....
instead of
numinternal(...)= n+ \sum_ ....

4)
b) 
typo:
"nonnegative" instead of "nonegative"


e)
The equation is kind of trivial because
\binom{n+t+s}{s}=\binom{n+t+s}{n+t}
and 
\binom{n+t}{t}= \binom{n+t}{n}


I wonder if this can be changed into something like this

\binom{n+ts}{s}\binom{n+t}{t}
=\binom{n+t+s}{t+s}\binom{t+s}{t}


so that the two sides are not products of the some pair of numbers.