From the week of September 25 to September 29 we chose Monday's problem (that you can find here):
Rosa is about to go on a long road trip and wants to take some books with her. Her mom told her that she can only bring $4$ books. Rosa has $5$ Science Fiction books, $3$ Thrillers, and $10$ Romantic Comedy books. If Rosa wants to bring at least one book from each category, but no more than $2$ from each, in how many ways can she select the books she will bring to the road trip?
21% of the students who tried the problem got it right on their first attempt.
The second most popular answer was exactly two times the correct answer. What might have happened?
In this problem we needed to count the number of ways to choose the books. However, the order in which the books are chosen does not matter! The only important thing is to make sure to choose the right amount of books from each category.
Those of you who got exactly two times the right answer probably chose the books in certain order. This is particularly important when choosing $2$ (or more) books from the same category. Think of it this way: It is the same to choose "The Day Earth Stood Still & The Ender's Game" as it is to choose "Ender's Game & The Day Earth Stood Still" from the Science Fiction category. Note the difference between:
- 2 Science Fiction books
- 1 Thriller book
- 1 Romantic Comedy book
- Science Fiction book 1
- Science Fiction book 2
- Thriller book
- Romantic Comedy book
In both we would choose $4$ books in the end; $2$ from Science Fiction, $1$ Thriller, and $1$ Romantic Comedy, but in the second method of counting it matters which Science Fiction was chosen first.
What do you think can be done to fix overcounting like this? Share your comments and questions below!