Jeffrey wants to start his second hour with a hard-rock song and finish with a pop classic. How many different playlists can Jeffrey prepare if he has chosen 4 hard rock songs, 5 soul pieces and 3 pop classics?
He play all 12 songs.
To calculate the total number of different playlists Jeffrey can prepare, we need to consider the arrangement of the songs.
First, let's select a hard-rock song for the starting position. Jeffrey has 4 choices for this.
Next, let's choose a pop classic song for the ending position. Jeffrey has 3 choices for this.
Now, for the remaining songs in the playlist, we have a combination of soul pieces to choose from. However, since the order of the soul pieces does not matter, we need to use combinations instead of permutations.
Jeffrey has 5 soul pieces to choose from, and he needs to select the remaining number of songs in the playlist. Since the playlist contains 4 hard-rock songs at the beginning and 1 pop classic song at the end, the number of remaining songs is the total number of songs minus the number of hard-rock songs and pop classic songs. So, the number of remaining songs is:
Total number of songs = Number of hard-rock songs + Number of soul pieces + Number of pop classic songs
Total number of songs = 4 + 5 + 3 = 12
Number of remaining songs = Total number of songs - Number of hard-rock songs - Number of pop classic songs
Number of remaining songs = 12 - 4 - 1 = 7
Now, we use combinations to select the soul pieces for the remaining positions. The number of combinations is:
Number of combinations = (Number of soul pieces + Number of remaining songs - 1) choose (Number of remaining songs)
Number of combinations = (5 + 7 - 1) choose 7
Calculating this combination using the formula:
(Number of combinations) = (5 + 7 - 1)! / (7! * (5 - 1)!)
(Number of combinations) = 11! / (7! * 4!)
Simplifying the calculation:
(Number of combinations) = (11 * 10 * 9 * 8) / (4 * 3 * 2 * 1)
(Number of combinations) = 11 * 10 * 3
Therefore, the total number of different playlists that Jeffrey can prepare is:
Total number of playlists = Number of choices for the starting position * Number of choices for the ending position * Number of combinations for the remaining positions
Total number of playlists = 4 * 3 * (11 * 10 * 3)
Total number of playlists = 4 * 3 * 330
Total number of playlists = 3960
So, Jeffrey can prepare a total of 3960 different playlists.