Task: In the library, shelves are filled with various books of different genres. If the library has $6$ detective books, $5$ sci-fi books, and $4$ novels, in how many ways can we choose $3$ books such that at least two books are of different genres?
Hana’s solution:
(1) Hana first calculated the number of ways to choose three books without regard to genre. There are a total of $15$ books in the shelves. She determined the number of ways as the number of $3$-permutations from the set of $15$ books, i.e., $$15\cdot14\cdot13=2\,730.$$
(2) However, she realized that this number also includes choices where all three books are of the same genre.
- (a) For detective books, of which there are a total of $6$ in the shelves, there are ${6\choose 3}=20$ such choices.
- (b) For sci-fi books, with a total $5$ books, there are ${5\choose3}=10$ choices.
- (c) For novels, numbering $4$ on the shelves, there are ${4\choose3}=4$ ways to choose three.
(3) Hana subtracted the selections containing books of the same genre from the total number of choices for trios of books, thus obtaining the number of ways to choose three books such that at least $2$ are of different genres: $$2\,730-20-10-4=2\,696$$ Did Hana make a mistake in her reasoning? If yes, find it!
Hana made a mistake in step (1). When choosing books, the order does not matter. Correctly, she should have determined the number of ways to choose three books as the number of $3$-combinations from $15$ books, i.e., ${15\choose 3}=455$. Thus, the resulting number of ways is: $$455-20-10-4=421$$
Hana made the first mistake in step (2a). The number of choices of three books of the detective genre should have been calculated as the number of $3$-permutations from $6$ books, i.e., $6\cdot5\cdot4=120$. She should have used $k$-permutations in steps (2b) and (2c) as well. Thus, the resulting number of ways should be: $$2\,730-120-60-24=2\,526$$
Hana made a mistake in step (3). She should have subtracted the product of the numbers of books of the same genre: $$2\,730-20\cdot10\cdot4=1\,930$$
Hana made no mistakes in her calculations.