Combinations with repetition - practice problems

Combinations with repetition, also called multicombinations, count the number of ways to select k items from n types where items can be repeated and order doesn't matter. The formula is C(n+k-1, k) = (n+k-1)!/[k!(n-1)!], derived using the "stars and bars" method. Unlike standard combinations, the same item can be chosen multiple times. For example, selecting 3 ice cream scoops from 5 flavors with repetition allowed gives C(5+3-1, 3) = C(7, 3) = 35 possibilities. Applications include distributing identical objects into distinct containers, polynomial coefficient problems, and inventory selection where stock is unlimited. This concept extends basic combinatorics to scenarios with replacement.

Task: Work through each problem with care and demonstrate your solution process for each one.