The Permutations Triangle

During the past few years it has become popular to explore Pascal's triangle by a discovery approach similar to that which George Polya presents in Mathematical Discovery (1960), especially pages 70-73 and 83-84. (The Bibliography at the end of this article will give other references indicating the rich variety of relationships found in this triangle.) The simple beauty of these relationships can be used to experience the joy of discovery and at the same time learn an ever more general method of problem solving. Some teachers have already begun incorporating such material into the high school curriculum, and I believe this practice should be increased because of its motivational value. Furthermore, the student who has derived enjoyment and profit from discovering the relationships of Pascal's triangle needs an opportunity to expand his work in this area. This article presents what I believe to be a natural extension suitable to the senior high school curriculum.