[MUSIC] So we want to find the number of k element model sets of the sets with n elements or which is the same number of putting k balls inside n boxes. To find this number, let us reinterpret this problem in a slightly different way. Suppose we have such a configuration of balls and boxes. Let us represent it as follows by a sequence of defined objects while circles will represent balls and vertical bars, which will separate boxes. So the space to the left of the first bar is the first box, then suppose we're going from left to right. And at first, we're not looking at the first bar bounding the lead box from the left and we see one ball inside the first box. And then we'll put a bar which separates the first and the second boxes and then we put another, there are no balls inside the second box and we put a separator between the second and the third box. So, we'll put a bar here an then there are three balls inside the third box. And then there is a bar separator between the third and the fourth box. And finally, there is one ball in the last, the right most box. So we get a configuration of, k balls, And instead of n boxes, we get n minus 1 separator bars. So this, this picture innately determines the configuration of balls inside boxes. So, we need to compute the number of such configurations and this is the standard problem involving binomial coefficients. Namely, we have k plus n-1 objects called balls or bars. And out of these, we need to select k which corresponds to balls. Or equivalently, we need to select n-1 corresponding to bars. So, configurations, Of balls in boxes. Correspond to configuration of, Bars and balls. And there are k+n-1 objects. And out of these, k are balls and n-1 are bars. So, the number of such configurations is, k+n-1 choose k, out of k+n-1 objects. We need to pick those which correspond to balls. We get a theorem. Theorem, k balls can be put, Into n boxes, In n+k-1, choose k, ways. Equivalently, The number of k-element multisets. In an n-element set is n+k-1, choose k. And right now, we will discuss another version of this problem about compositions of integer number into positive integer assignments. [MUSIC]