Victor is picking out some movies to rent, and he is primarily interested in foreign films and horror films.  He has narrowed down his selections to 9 foreign films and 6 horror films.
Step 2 of 2 :  
How many different combinations of 4 movies can he rent if he wants at least two foreign films?

Victor has 9 foreign films and 6 horror films to choose from, and he wants to rent 4 movies with at least 2 foreign films. We can break this into cases:

### Case 1: 2 foreign films and 2 horror films
- The number of ways to choose 2 foreign films from 9:  
  \[
  \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36
  \]
- The number of ways to choose 2 horror films from 6:  
  \[
  \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15
  \]
- Total combinations for this case:
  \[
  36 \times 15 = 540
  \]

### Case 2: 3 foreign films and 1 horror film
- The number of ways to choose 3 foreign films from 9:  
  \[
  \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
  \]
- The number of ways to choose 1 horror film from 6:  
  \[
  \binom{6}{1} = 6
  \]
- Total combinations for this case:
  \[
  84 \times 6 = 504
  \]

### Case 3: 4 foreign films and 0 horror films
- The number of ways to choose 4 foreign films from 9:  
  \[
  \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
  \]
- Total combinations for this case:
  \[
  126
  \]

### Total number of combinations
Summing the results from all cases:
\[
540 + 504 + 126 = 1170
\]

Thus, the total number of different combinations of 4 movies he can rent, with at least 2 foreign films, is **1170**.

Would you like further details or have any questions?

### Related Questions:
1. How would the number change if he needed at least 3 foreign films?
2. What if there were 10 foreign films instead of 9?
3. How would the combination change if he wanted exactly 2 foreign films?
4. What if Victor could rent 5 movies instead of 4?
5. How would the number of combinations change if he wanted only 1 horror film?

**Tip:** When solving problems involving combinations, breaking the problem into cases based on specific conditions (like "at least" or "exactly") can help simplify the solution.

Victor is picking out some movies to rent, and he is primarily interested in foreign films and horror films. He has narrowed down his selections to 9 foreign films and 6 horror films. How many different combinations of 4 movies can he rent if he wants at least two foreign films?

Learn how to calculate the number of ways Victor can rent 4 movies when choosing from 9 foreign films and 6 horror films, ensuring at least 2 foreign films are included. This problem explores combinatorics using combination formulas and binomial theorems. Suitable for students in Grades 10-12.

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