Question

This question has been posted before (a good while ago so it's not likely anyone will respond to comments), but the answers are either wrong or don't actually explain how the answers are obtained. (The answers in the back of the book are 24 for 3 and 100 for 5.)

Mathematicians at the University of Florida solved a 30-year-old math problem using the theory of partitions. (Explore, Fall 2000.) In math terminology, a partition is a representation of an integer as a sum of positive integers. (For example, the number 3 has three possible partitions: 3, 2 + 1 and 1 + 1 + 1.) The researchers solved the problem by using “colored partitions” of a number, where the colors correspond to the four suits—red hearts, red diamonds, black spades, and black clubs—in a standard 52-card bridge deck. Consider forming colored partitions of an integer.

1. How many colored partitions of the number 3 are possible? (Hint: One partition is 3 of hearts; another is 2 of diamonds  + 1 of clubs.)

2. How many colored partitions of the number 5 are possible?

Answer 1