Final answer:
By treating the five consecutive rap songs as one entity along with the six rock songs, the permutations of seven items (7!) are considered for arrangement. Since the rap songs themselves have an internal ordering, their permutations (5!) are also calculated, leading to a total of 7! times 5! arrangements.
Step-by-step explanation:
The question involves calculating permutations where a condition is imposed on the order in which items are arranged. In this case, we have a total of 11 songs, out of which 5 rap songs must be played consecutively. To solve this problem, we can treat the 5 rap songs as a single entity since they are to be played consecutively. This reduces the problem to arranging 6 rock songs and 1 block of rap songs, effectively 7 items in total.
The permutations of 7 items are 7!, which is equal to 7 × 6 × 5 × 4 × 3 × 2 × 1. However, since the 5 rap songs can also be rearranged amongst themselves, we must also consider the permutations of these 5 songs, which is 5!, or 5 × 4 × 3 × 2 × 1. Therefore, the total number of different orders the songs can be played in is 7! × 5!, which can be calculated as follows:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
5! = 5 × 4 × 3 × 2 × 1 = 120
Therefore, the total number of different orders is 5,040 × 120 = 604,800 possible arrangements.