Many counting problems are solved by establishing a bijection between the set to be counted and some easy-to-count set. This kind of proofs are usually called (non-rigorously) combinatorial proofs.
The number of k-compositions of n is equal to the number of solutions to 
in positive integers.
count the number of solutions to in nonnegative integers We call such a solution a weak k-composition of n.
Formally, a multiset M on a set S is a function 
. For any element 
, the integer 
is the number of repetitions of x in M, called the multiplicity of x. The sum of multiplicities 
is called the cardinality of M and is denoted as | M | .
we have already evaluated the number 
. If 
, let zi = m(xi), then is the number of solutions to 
in nonnegative integers, which is the number of weak n-compositions of k, which we have seen is 
.
We can think of it as that n labeled balls are assigned to m labeled bins, and 
is the number of assignments such that the i-th bin has ai balls in it.