Help needed quickly with clear, thorough explanations.

Background: The introduction to Chapter 8 describes an innocuous counting problem. A bakery sells 6 different varieties of cupcakes (chocolate, vanilla, red velvet, etc.). How many ways are there to fill a box with 24 cupcakes from the 6 varieties? The order in which the cupcakes are selected is unimportant; all that matters is the number of each variety in the box after they are chosen. The techniques needed to solve this problem aren't discussed until $8.9, when the book claims (without explanation) that "the number of ways to place n indistinguishable balls into & bins is k_1 ). The goal of this exploration is to prove this statement. The Original Problem [Problem A]: Suppose you are at a cupcake shop that has k varieties of cupcakes. You will fill a box with n cupcakes for Casper's birthday party. Being indecisive, before you can make your order, you must determine the number of ways you can fill the box with n cupcakes. 1. Show that this problem [Problem A] is equivalent to the following one: [Problem B ] Given positive integers n and k, how many ways are there to write n as a sum of k integers 21, 2, . .., Ix with 0