A Generalization of Placing Identical Items into Identical Bins

A common approach to counting the number of ways to place identical items into identical bins is by casework. In this article, an alternative approach is introduced and robust mathematical formulas are established to calculate the number of ways of placing arbitrary number of identical items into arbitrary number of identical bins. Firstly, single closed formulas for the cases of two and three bins are developed for arbitrary number of items. Secondly, a recursive formula for more than three bins is derived for arbitrary number of items. This recursive formula reduces the number of bins by one in each step until reaching the base case of three bins for which the closed formula derived in this paper can be applied. A Python program is implemented using the derived formulas that can count the number of ways for arbitrary bins and items.