A study of a dice problem

Abstract

Dice are well-studied objects in combinatorial studies. In this paper, we study the following problem about dice: Given m dice, each of which has n faces, how many arrangements of those dice are there such that the sum of their outcomes equals to p? We will give a computer solution as well as a combinatorial analysis of this problem. After studying the relationship between this dice problem and multinomial coefficient, we will further generalize this dice problem to its extreme and provide corresponding solutions.