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: total number of songs is 5!x6!x3! Number of different albums is 5x6x14C3 Is this correct. Thanks in advance.
Your solution is not correct. Let me explain how you can solve this problem.
To find the number of different albums that can be formed, we can use the principle of combinations. We need to consider two different cases:
Case 1: Exactly one rock song and one country song are chosen.
In this case, we can choose one rock song out of the 5 available rock songs in 5 ways, and one country song out of the 6 available country songs in 6 ways. For the remaining song(s), we can choose any combination of songs from hip-hop, rock, or country, so we have 14 songs to choose from. Therefore, the number of albums in this case is 5 * 6 * 14C3.
Case 2: More than one rock song and/or more than one country song are chosen.
In this case, we can choose multiple rock songs and/or multiple country songs. The number of options for choosing the rock songs can vary from 2 to 5, and similarly for the country songs it can vary from 2 to 6. So, we need to sum up the number of albums for each possible combination.
1 rock song and 1 country song: 5 * 6 * 14C3 (as calculated in Case 1)
2 rock songs and 1 country song: 5C2 * 6 * 14C2
3 rock songs and 1 country song: 5C3 * 6 * 14C1
4 rock songs and 1 country song: 5C4 * 6 * 14C0
1 rock song and 2 country songs: 5 * 6C2 * 14C2
1 rock song and 3 country songs: 5 * 6C3 * 14C1
1 rock song and 4 country songs: 5 * 6C4 * 14C0
To get the total number of albums, we add up the number of albums in each case:
Total number of albums = (5 * 6 * 14C3) + (5C2 * 6 * 14C2) + (5C3 * 6 * 14C1) + (5C4 * 6 * 14C0) + (5 * 6C2 * 14C2) + (5 * 6C3 * 14C1) + (5 * 6C4 * 14C0)
By calculating the values of the combinations, you can find the final answer.