Roxie is picking out some movies to rent, and she is primarily interested in horror films and foreign films.  She has narrowed down her selections to 20 horror films and 8 foreign films.
How many different combinations of 3 movies can she rent if she wants at least two horror films?

Roxie has 20 horror films and 8 foreign films available to rent, and she wants to rent 3 movies in total with at least 2 being horror films. We can break this down into two cases:

### Case 1: She rents 2 horror films and 1 foreign film.
- There are \( \binom{20}{2} \) ways to choose 2 horror films from 20.
- There are \( \binom{8}{1} \) ways to choose 1 foreign film from 8.

So, the total number of combinations for this case is:
\[
\binom{20}{2} \times \binom{8}{1} = \frac{20 \times 19}{2} \times 8 = 190 \times 8 = 1520
\]

### Case 2: She rents 3 horror films.
- There are \( \binom{20}{3} \) ways to choose 3 horror films from 20.

The total number of combinations for this case is:
\[
\binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
\]

### Total combinations:
Now, we add the results from both cases:
\[
1520 \, (\text{Case 1}) + 1140 \, (\text{Case 2}) = 2660
\]

Thus, the total number of different combinations of 3 movies Roxie can rent is **2660**.

Do you want further details or have any questions?

Here are five related questions to explore:
1. How would the result change if Roxie could only rent 2 movies?
2. What if she wanted at least one foreign film instead?
3. How many combinations would there be if Roxie could rent any 3 movies with no restrictions?
4. How would the number of combinations change if there were 10 foreign films instead of 8?
5. What if Roxie had to rent exactly 1 horror film and 2 foreign films?

**Tip:** In combinatorics problems, always consider if there are distinct cases (like different numbers of horror/foreign films) and calculate them separately.

Roxie is picking out some movies to rent, and she is primarily interested in horror films and foreign films. She has narrowed down her selections to 20 horror films and 8 foreign films. How many different combinations of 3 movies can she rent if she wants at least two horror films?

Solve a combinatorics problem where Roxie picks 3 movies from a selection of 20 horror films and 8 foreign films. Learn how to calculate combinations with at least 2 horror films using the combination formula. This problem involves counting methods suitable for Grades 10-12.

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