combinations and proability There are 120 books on 24 topics with 5 volumes on each topic.
What is the probability of choosing 3 books such that they all belong to different topics?
What is the probability of choosing "r" books such that at least one book is on a repeated topic?
 A: What is the probability of choosing 3 books such that they all belong to different topics?
As you wrote in your comment, there are a total of ${120 \choose 3}$ ways to pick 3 books. There are ${24 \choose 3}$ ways to pick 3 distinct topics, and $5^3$ ways to pick from the books within those topics. Therefore, we get: $$\frac{{24 \choose 3} \cdot 5^3}{{120 \choose 3}}$$
What is the probability of choosing "r" books such that at least one book is on a repeated topic?
This calls for complementary counting. Let's find the number of ways to choose "r" books such that no books are of the same topic. Similar to above, we calculate a total of ${120 \choose r}$ ways to pick r books. There are ${24 \choose r}$ ways to pick r distinct topics, and $5^r$ ways to pick the books from said topics. We get: $$\frac{{24 \choose r} \cdot 5^r}{{120 \choose r}}$$
Since we're doing complementary counting, we have to subtract that probability from 1: $$1 - \frac{{24 \choose r} \cdot 5^r}{{120 \choose r}}$$
Hope this helped.
