Combinatoric Identity for a Balls and Boxes Problem

John continues the "balls and boxes" series by answering a question a student posed : If n indistinguishable balls are placed into alpha distinguishable boxes with at least one ball per box, how many arrangements are possible if n is allowed to vary from alpha to k? This problem is equivalent to the number of integer solutions of the equation x_1 + x_2 + x_3 + ... x_(alpha) less than or equal to n [n between alpha and k, inclusive]. John's solution has a nice "short cut" for "balls and boxes" (or ball and urn) problems where at least one ball is placed in each urn. As a bonus, John helps the student by simplifying using a combinatoric identity and proves it via a combinatoric argument (not algebra). Here are the original "balls and boxes" video https://youtu.be/1crNnZkIvac and another follow up video https://youtu.be/0bWyiAZOK8s . #combinatorics #combinations #ballandurn