Question

Ali is picking out some movies to rent, and she is primarily interested in action movies and dramas. She has narrowed down her selections to 10 action movies and 20 dramas. Step 2 of 2 : How many different combinations of 4 movies can she rent if she wants at least two action movies

Answer 1

Answer:

We conclude that exists 11160 different combinations.

Step-by-step explanation:

We know that Ali has narrowed down her selections to 10 action movies and 20 dramas.

First case:  

Out of 10 action films she chooses 2 and out of 20 drama films she chooses 2. We get

$C_2^{10}\cdot C_2^{20}=\frac{10!}{2!(10-2)!}\cdot \frac{20!}{2!(20-2)!}=45\cdot 190=8550$

Second case:  

Out of 10 action films she chooses 3 and out of 20 drama films she chooses 1. We get

$C_3^{10}\cdot C_1^{20}=\frac{10!}{3!(10-3)!}\cdot \frac{20!}{1!(20-1)!}=120\cdot 20=2400$

Third case:  

Out of 10 action films she chooses 4 action films. We get

$C_4^{10}=\frac{10!}{4!(10-4)!}=210$

Threfore, we get:

8550+2400+210=11160

We conclude that exists 11160 different combinations.