Number of ways to pick 4 books from different genres There are 4 genres, each genre has 11 books. I want to choose 4 books such that (order does not matter):


*

*all books are from the same genre

*all books are from different genres

*the books are from exactly three genres

*the books are from exactly two genres


The following are my attempts:


*

*$11 \choose 4$ chooses 4 books from one genre that contains 11 books. There are 4 genres. Therefore, $4 \times {11 \choose 4}$ total combinations.

*$11^4$ total combinations. 11 possibilities for the first, 11 for the second, 11 for the third, 11 for the fourth. Does this take order into account? Again, I want order to be irrelevant.

*$11^3$ for three slots, then $10 \times 3$ for the remaining slot. $11^3 \times 30$ total combinations.

*${11 \choose 2} \times {11 \choose 2} \times {4 \choose 2}$. Since you are choosing 2 books from 11 twice, and you are choosing 2 genres from 4 genres.
Any advice on these attempts would be appreciated.
 A: *

*Correct

*Correct (If you multiplied by a factor of $4!$ then order would matter, but the way you have counted says order does not matter)

*Not quite. Your method of counting says choose a book from each of 3 genres (but which ones?), then pick one of the genres and choose a second book from it. There are two problems with this. First, you haven't said which 3 genres are being used, so you need to multiply by a factor of $\binom{4}{3}$. Second, the order in which you choose the two books from one genre matters in your method of counting. Suppose the books are $A_1,\dots A_{11}$ for the first genre, $B_1,\dots,B_{11}$ for the second, and so on. If I choose $A_1,B_1$, and $C_1$ and then pick genre $A$ and choose $A_2$, that is counted differently from choosing $A_2$, $B_1$, and $C_1$ and then picking genre $A$ and choosing $A_1$. So you need to divide by a factor of $2$. Another way to think about this is: choose 3 genres ($\binom{4}{3}$), choose one of them to have two books ($3$), choose one book each from two genres ($11^2$), and choose two books from the third ($\binom{11}{2}$). 

*Almost. This would be correct if the question said two books are from one genre and two books are from another. But what if 3 are from one and 1 is from another?

