Ways to Pick Shirts from a Set

How many ways can we pick n shirts out of four different shirts? Provide answers for n = 0, 1, 2, 3, 4, 5.

This problem is a classic example in the area of combinatorics, where we deal with counting the number of ways to choose items from a set. When you are asked to pick n shirts out of four, the task is fundamentally related to the concept of combinations. Combinatorics often involves understanding the structure and counting possibilities of a given situation. In this problem, each shirt represents a unique item, and picking n out of those available provides a clear application of the concept of combinations without repetition.

For instance, when solving this type of problem, it's important to distinguish between combinations and permutations. Combinations focus on selecting items where the order does not matter, unlike permutations where the sequence is crucial. Here, since we're merely selecting shirts with no concern for order, combinations are appropriate. The standard formula for combinations involves factorials to account for selection without respect to order.

Additionally, evaluating this problem requires us to address different values of n. For small values, such as n = 0 or n = 5, we reach into edge cases where the solution highlights principles such as the meaning of choosing "nothing" at all, yielding exactly one way to choose zero items, and the constraint of choosing all available items. Overall, this problem teaches an essential principle in discrete mathematics: making structured selections from finite sets.