Six rock songs and four rap songs are on a disc jockey's playlist for a radio show. How many different orders can the songs be played in if the following conditions apply? Show your work!
(a)
there are no restrictions on the order
(b)
all the rap songs are played consecutively
(a) Since there are 10 songs in total (6 rock songs and 4 rap songs), the number of different orders they can be played in is 10! (10 factorial), which is equal to 3,628,800.
(b) Since all the rap songs must be played consecutively, treat the 4 rap songs as a single unit. This reduces the number of items to be arranged to 7 (5 units of rock songs + 1 unit of rap songs). The number of different orders they can be played in is then 7! (7 factorial), which is equal to 5,040.
are you sure for b?
I apologize for the error in my previous response for part (b). Let me correct it:
For part (b), since all the rap songs must be played consecutively, treat the 4 rap songs as a single unit. This reduces the number of items to be arranged to 7 (6 rock songs + 1 unit of rap songs). The number of different orders they can be played in is then 7! (7 factorial), which is equal to 5,040.
Thank you for pointing that out, and I apologize for any confusion.
you were still wrong on part B.
The answer is 120,960 orders.
Can you explain why?
I apologize for the error in my previous responses. Let's correct the calculation for part (b) where all the rap songs are played consecutively.
We have 6 rock songs and 4 rap songs. Taking the 4 rap songs as a single unit, we have 6 + 1 = 7 items to arrange.
The total number of orders in which the 7 items can be arranged is 7! (7 factorial) because although the rap songs must stay together, they can still be in different orders within that unit.
So, the correct number of different orders is 7! = 5040.
To correct the error in my responses, here is the updated calculation:
7! = 5040
Thank you for pointing out the mistake, and I appreciate your patience.
No. The answer is not 7!.
The answer is 120,960.
I appreciate your understanding. Let me explain why the correct answer for part (b) is 120,960:
When all the rap songs are played consecutively, we can treat the 4 rap songs as a single unit. This reduces the number of items to be arranged to 6 (the 5 rock songs and the unit of 4 rap songs). The number of different orders they can be played in is then 6! (6 factorial).
Calculating 6!, we get:
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
However, within the unit of rap songs, there are 4 songs that can be arranged in 4! ways.
So, the number of different orders when all the rap songs are played consecutively is:
6! x 4! = 720 x 24 = 17,280
Therefore, the correct answer for part (b) is 17,280.
I apologize for the mistake in my previous responses. Thank you for your patience and for pointing that out.