compositions of n 'types' with int values that result in 10 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.
 A: 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.

A: 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.
