Problem 1: Stars and Bars

Imagine that you have a collection of \(n\) indistinguishable stars:

☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆

Above, we have \(n = 10\) stars. Suppose that we want to create \(k\) distinguished buckets of stars where each bucket must have at least one star in it. For example, if \(k = 3\), we can create three buckets from the 10 stars as follows:

☆ ☆ ☆ ☆
☆ ☆ ☆
☆ ☆ ☆

The first bucket has 4 stars, the second and third, 3 stars each. Another option would be:

☆ ☆
☆ ☆ ☆
☆ ☆
☆ ☆ ☆

Because the buckets are distinguished, the following collection of buckets is different from the previous collection of buckets:

☆ ☆
☆ ☆
☆ ☆ ☆
☆ ☆ ☆

Give a combinatorial description for the number of ways we can create \(k\) distinguished buckets from \(n\) indistinguishable stars. Justify your combinatorial description in a sentence or two.

Problem 2: Sums

With the previous problem in mind, give a combinatorial description for the number of ways of creating \(k\)-tuples positive integers whose sum is equal to \(n\). Justify your combinatorial description in a sentence or two.

Problem 3: More

Describe in a few sentences each two additional real-world situations where you can directly apply the result from problem 1. You should describe the situation and how the result of problem 1 counts the number of objects in the situation.

Problem 4: Generalizing

Extend the result from problem 1 so that you allow for a bucket to be potentially empty. How do you then generalize the situations from problems 2 and 3 so that your extended result applies?