The number of compositions of a number n into k parts is given by the binomial coefficent ${n-1 \choose k-1}$.
Is there a closed formula to this question, when the summands of the composition are limited to a range [1;m] where m is an arbitrary number strictly smaller than n?
Example: Compositions of n=6 into k=3 parts with m = 3
(2,2,2),(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)
gives us 7 different compositions. The unrestricted composition has size
${5 \choose 2} = 10$
I tried this out for multiple combinations of numbers and I could not figure out a rule or a pattern that held for every combination of n,k and m.