number of elements in subset sum problem I have a set of numbers $i=1,...,100$.
How many combinations exist using numbers from this set that sum to 100 of length 8?
So for example these would be valid solutions:
$(1, 2, 3, 4, 5, 6, 7, 72)$,
$(10,11,22,1,5,8,9,34)$
also, the order is important, that means
$(1, 2, 3, 4, 5, 6, 7, 72) \neq (2,1, 3, 4, 5, 6, 7, 72)$
also zeros can be added in any amount, e.g.
$(100,0,0,0,0,0,0,0)$.
numbers can also be used more than one time, e.g.
$(10,10,10,10,10,10,10,30)$

If $(100,0,0,0,0,0,0,0) \neq (0,100,0,0,0,0,0,0)$, for instance, you're looking for the number of solutions $(x_1, x_2, ..., x_8)$ to the equation $x_1 + ... + x_8 = 100$, where each $x_i$ is a non-negative integer.
The solution is given by a stars-and-bars argument.  In your particular case, the answer is
$$ \binom{107}{7}. $$
Otherwise, if the zeroes can only be added at the end, you're looking for the number of solutions to $x_1 + ... + x_k = 100$, where $k \le 8$ and each $x_i$ is a positive integer.  A similar stars-and-bars argument gives
$$ \sum_{k=1}^8 \binom{99}{k-1}. $$
