There are 5 rock songs, 6 country songs, and 3 hip-hop songs. How many different albums can be formed using the songs if the album should contain at least 1 rock song and 1 country song?
My solution: 14C5x14C6x14C3-5C1x6C1
Is this correct? Thanks in advance
To find the number of different albums that can be formed, we can take into account the number of ways we can choose the rock, country, and hip-hop songs.
First, let's calculate the number of ways to choose the songs without any restrictions. We can do this by taking the product of the combinations of each type of song:
Number of ways to choose all rock songs: 14C5
Number of ways to choose all country songs: 14C6
Number of ways to choose all hip-hop songs: 14C3
Next, we need to subtract the number of albums that don't have at least 1 rock song or 1 country song. We can do this by calculating the number of albums with only country and hip-hop songs, and then subtracting it from the total.
Number of ways to choose only country songs: 6C1
Number of ways to choose only hip-hop songs: 3C1
Now, we subtract this from the total number of combinations:
Total number of albums = (14C5) * (14C6) * (14C3) - (6C1) * (3C1)
Calculating this expression will give you the correct number of different albums that can be formed.
Please note that your solution, 14C5 * 14C6 * 14C3 - 5C1 * 6C1, is almost correct. However, the subtraction part should use 6C1 instead of 5C1 since we are considering all country songs.