In how many ways can 15 identical computer science books and 10 identical psychology books be distributed among five students?
So I'm trying to figure this out: I know how to calculate 15 identical cs books: C(15+6-1, 6-1) and also 10 identical psych books: C(10+6-1, 6-1), but I do not know how to consider the combinations with both books.
By the way, I've asked a few questions on here in the past hour; I just wanted to say that these aren't homework problems, but I'm doing these problems to study for a midterm tomorrow. I guess there isn't any way to prove that….but just wanted to put it out there.
Thanks for your help!
Best Answer
You can consider it two distributions in succession: first you distribute the $15$ computer science books, and then you distribute the $10$ psychology books. Thus, the final answer is the product of the number of ways of making each of these distributions. However, your calculations for those numbers are a bit off. The number of ways of distributing $n$ identical objects to $k$ distinguishable bins is $$\binom{n+k-1}{k-1}=\binom{n+k-1}n\;.$$
Thus, the computer science books can be distributed in $$\binom{15+5-1}{5-1}=\binom{19}4$$ ways, and the psychology books in $$\binom{10+5-1}{5-1}=\binom{14}4$$ ways, and the final answer is $$\binom{19}4\binom{14}4\;.$$