Compositions: distribution of indistinguishable objects

How to make a model of the distribution of indistinguishable objects in two distinct boxes. Well, let’s see with an example. We have 7 identical objects, and we have to put them into 4 boxes labelled from 1 to 4. Well, let’s try, and then we finish the distribution. We have 2 elements in the box number 1, 0 elements in box number 2, 2 elements in box number 3, and 2 elements in box number 4. It is enough to know these numbers, 2, 0, 3, 2 to identify perfectly the distribution of these 7 objects. What do you obtain is a sequence of numbers, namely 2, 0, 3, 2, whose sum is equal to 7.

2 corresponds to the number of elements that have been put in the first box, 0 in the second, 3 and the third, and 2 in the fourth. And such a sequence is called a composition, namely a full composition of 7. Marginally, if you have two integers, k and n, at n composition of k is the end sequence k1 kn of natural numbers, whose sum is k. We speak of a integer solution of the equation k1 plus plus kn equal to k. So this ki are positive, or natural numbers, more precisely, and that sum is k. Order counts.

For instance, the sequence 2, 0, 3, 2 is different from the sequence 3, 0, 2, 1, because in the first case, we put two elements in the first box, 0 in the second, 3 in the third, 2 in the fourth, whereas in the second, we put 3 elements in the first box, 0 in the second, 2 elements in box number 3, and number 4.