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Let's say I have $8$ different 'types' that each can have a count assigned of $0-10$.

I want to know how many compositions there are so that the sum of counts is always 10.

Example: a=> (10) and b through h=> (0) is valid, but a=> (10) and b=> (10) is not.

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    It would probably be easier to work with assigning integers $0$ to $10$ to these eight unknowns, to total $10$. – 2017-02-02
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    A better word than "combinations" here is [compositions](https://en.wikipedia.org/wiki/Composition_(combinatorics)) because the types are distinguished (which amounts to treating the order of assignments as important). – 2017-02-02
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    Switched to integers. 'Compositions' sounds about right, thanks. – 2017-02-02

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Counting the possible assignments is discussed in many problems here with the "stars and bars" technique.

Consider ten stars and seven (one less than eight) bars arranged in a line. There are $\binom{17}{7} $ possibilities.

Interpret the bars as separating the stars into eight summands from left to right corresponding to the eight types.

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    Thanks, besides your Wiki reference, this article is also quite good for understanding: https://en.wikipedia.org/wiki/Multiset#Counting_multisets – 2017-02-02
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This is equivalent to the number of ways of partitioning the number $10$ into integers. (Where we associate the value $n\times10\%$ to each integer $n$ to return to your original problem).

There are several possible answer depending on whether you allow for repeatition, and whether you include the number $0$:

  • Ways of partitioning the number $10$ into distinct sums of integers $1\ldots 10$ is $P_{10}=42$ allowing for repetition. See here.
  • Ways of partitioning the number $10$ into distinct sums of integers $0\ldots 10$ is infinite if we allow for repetition.
  • Ways of partitioning the number $10$ into distinct sums of integers $1\ldots 10$ is $Q_{10}=10$ when we do not allow repetition. See here.
  • Ways of partitioning the number $10$ into distinct sums of integers $0\ldots 10$ is $20$ when we do not allow repetition.
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    I don't think that answers my question. Variables a through h have a value between 0 and 10 (both included). The sum of all eight variables needs to be 10 (100% in my case). How many combinations are there that lead to that sum of 10. The variables themselves can't repeat, their respective values can. – 2017-02-02
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    These,are not partitions, but (weak) compositions of ten into eight summands. See the link in my Comment on the Question. – 2017-02-02